What is the basis for the LDS belief that religious truth is confirmed by what you feel?

I know this is late and there are already lots of answers. But I'll just add one thing.

If salvation was dependent on believing in the bible after being convinced by historical evidence of its truthfulness, and then correctly interpreting the Bible in order to correctly follow it, that would mean that Salvation is dependent on a persons access to information and their intelligence.

If someone knows Greek and Hebrew, well then they must have an easier time interpreting the Bible and thus they must have an easier time knowing the path to Heaven, simply due to their intelligence.

Whearas Mormons, and many other Christians as well, would believe that a belief in Christ originates from faith. Faith leads to a spiritual manifestation, and a spiritual manifestation then constitutes personal evidence upon which a person can build a testimony of Jesus Christ and the interpretation of His Gospel.

Or in other words, the ability to know the path to Salvation has nothing to do with a persons intelligence and their access to information and has everything to do with Faith and the Holy Spirit. And as many of these answers have shone, the Holy Ghost speaks through feelings.

So if some guy has access to dead sea scroll information while another guy doesn't, neither has a better or worse chance at getting to heaven.

The teaching that religious truth is confirmed by feeling has support from several LDS scriptures. In the Book of Mormon, Nephi tells his brothers that they had heard the voice of God several times:

1 Ne 17:35

" ... ye have heard his voice from time to time; and he hath spoken to you in a still small voice, but ye were past feeling, that you could not feel his words...

The voice of God was described on a couple of occasions. One of them is:

Helaman 5:30

And it came to pass when they heard this voice, and beheld that it was not a voice of thunder, neither was it a voice of a great tumultuous noise, but behold, it was a still voice of perfect mildness, as if it had been a whisper, and it did pierce even to the very soul--.

A different prophet began a discourse on acquiring faith thus:

Alma 32:28

"Now we will compare the word unto a seed. Now, if ye give place, that a seed be planted in your heart, behold, if it be a true sees, or a good sees, if ye do not cast it out by your unbelief, that ye will resist the Spirit of the Lord, behold it will begin to swell within you breasts; and when ye feel these swilling motions, ye will begin to say within yourselves--it must needs be that this is a good seed, or that the word is good, for it beginneth to enlarge my soul; yea, it beginneth to enlighten my understanding, yea, it beginneth to be delicious unto me."

He expounds the process in more detail. LDS understand this discourse to explain the parable of the mustard seed in Matt. 13:31-32.

In the Doctrine and Covenants, Oliver Cowdery was told:

D&C 6:22-23

Verily, verily I say into you, if you desire a further witness, cast your mind upon the night that you cried unto me in your heart, that you might know concerning the truth of these things. Did I not speak peace to your mind concerning these things?

Also, when Oliver Cowdery attempted to take a turn translating the Book of Mormon and failed, the Lord explained:

D&C 9:8-10

Behold, you have not understood; you have supposed that I would give it unto you when you took no thought save it was to ask me. But behold, you must study it out in your mind; then you must ask me if it be right, and if it is right, I will cause that your bosom shall burn within you; therefore you shall feel that is is right. But if it be not right, you shall have no such feelings, but you shall have a stupor of thought which shall cause you to forget the thing which is wrong; therefore you cannot write that which is sacred save it be given you from me. Now, if you had known this you could have translated, nevertheless, it is not expedient that you should translate now.

The divinely inspired feeling which is described in these and related scriptures is not quite the same as any ordinary, common, human feeling, although it does resemble and may prompt many of the better kind.

Though not a Mormon, I have heard Elder Ballard's argument (i.e., a line of reasoning which turns into a logical and well supported argument for the legitimacy of a belief) in religious circles all my life. It is common, I suggest, to adherents of almost all of the world's religions. Feelings are important, whether in matters of religion, or romance, or relationships, or in any substantive and meaningful aspect of life. Even the most critical, unbiased, brilliant and empirically driven scientist--a virtual Mr. Spock--will have feelings about his or her hypotheses and discoveries, successes and failures. Not for naught do we depict a scientist who has just made a paradigm-shifting discovery, shouting ecstatically,

"EUREKA!"

Having said that, when it comes to matters of faith, emotions can often provide inner confirmation of the truth of one's beliefs. This can be a good thing, especially when this confirmation is confirmed, as it were, by other likeminded people who share our presuppositions. The Bible tells us,

"Where no counsel is, the people fall: but in the multitude of counsellors there is safety" (Proverbs 11:14 KJV).

And

"Without counsel purposes are disappointed: but in the multitude of counsellors they are established" (Proverbs 15:22 KJV).

And

"For by wise counsel thou shalt make thy war: and in multitude of counsellors there is safety" (Proverbs 24:6 KJV).

This confirmation, even when a "multitude of counselors" agree with you, can also be a less-than-good thing, particularly when the substance of those presuppositions and the process through which they became presuppositions are faulty. After all, there is such a thing as "group think," and in the midst of so much agreement, it is a very brave soul indeed who says,

"The emperor has no clothes!"

In religious matters, people are basically searching for truth. They may start out with an open mind, itself a good thing, but eventually through a very complicated and sometimes circuitous process they decide to close their minds when their heads and their hearts latch on to something.

A respected and world-famous Christian leader once said to me in a chance encounter many years ago (and I paraphrase):

"Having an open mind can be a good thing. Like the hungry person at a smorgasbord, however, keeping one's mouth open is not a good idea. Sooner or later, after surveying all the dishes before him or her, they need to chomp down on something if they want to satisfy their hunger!"

Similarly, humans find it natural to chomp down mentally, now and then, on a "keeper" (i.e., an idea, theory, proposition, or even a paradigm with which they are comfortable, at least for the time being). However, sometimes with some people a natural credulity leads them to be duped, preventing them from seeing things differently and maybe even more accurately and closer to the truth. Feelings can often prove to be hindrances to further reflection.

As Shakespeare said through his dramatic character Hamlet:

"Ay, there's the rub."

What then is a person to do?

A wise friend of mine who died a number of years ago said to me (and I paraphrase):

"The clincher, for me, which sort of settles the rightness of a belief, or a course of action, or the truthful answer to a perplexing question is an inner peace or quietude."

He would then quote the Bible verse found in Philippians, chapter 4:

"Be anxious for nothing, but in everything by prayer and supplication with thanksgiving, let your requests be made known to God. And the peace of God, which surpasses all comprehension, will guard your hearts and your minds in Christ Jesus" (vv.6-7, my italics).

There is peace with God, and then there is the peace of God. The former is a gracious gift from a loving God from whom we are estranged because of our sin. Once we receive forgiveness on the basis of Jesus' death, God no longer holds our sins against us (which he has every right to do) but sees us "in Christ" (see 2 Corinthians 5:17-19). Think of peace with God as the removal of enmity between God and us made possible through Jesus' death on our behalf.

The latter (i.e., the peace of God) is what we might call existential peace, the kind of peace which settles over us as we pray, supplicate, and make our requests known to God in a spirit of thankfulness and expectation. This peace could be described as having an inner assurance we know that we know." We may not be able to put our finger on why we are at peace in whatever circumstance we may be, because this kind of peace "surpasses comprehension."

This kind of peace comes to us through God's Holy Spirit, who bears witness with our spirits that since we are God's children, we can rest assured we are in good hands. This resting assured, according to the imagery of Philippians 4:6-7 is like a garrison of soldiers which guards us and keeps our hearts focused on the peace which only Jesus can give.

In conclusion, feelings can be both a blessing and a curse. When our feelings are in accord with the truth of God's word, they are a blessing. When our feelings are not in accord with the truth of God's word, they are a curse. Planting one's flag, or drawing a line in the sand, and saying, "This (fill in the blank) is what I believe because God's word gives me the inner peace it is so. To arrive at that transcendent feeling requires a step of faith, but then isn't that where feelings have their genesis? Feelings make a great cart, but they were never intended to take the lead, so to speak; that's the horse's job. These words of Jesus give us assurance in this regard:

"If anyone is willing to do . . . [the Father's] will, he will know of the teaching, whether it is of God or whether I speak from Myself" (John 7:17 NAS, my emphasis).

I listened to the provided video, and I feel you miss the key point:

You will never know a spiritual truth except it is revealed to you by the Holy Spirit. This is what Elder Ballard wants to say.

He is not saying that every feeling we get comes from God. He is also simplifying a little bit to get his point across.

You question of "What does it mean that religious truth is confirmed by what you feel?" is hard to answer, because what that means varies greatly. What he actually wants to say is: Physical Evidence will never be enough to produce faith or to find spiritual truth. Lack of physical evidence is no excuse for lack of faith, because God does communicate with us. But how this communication expresses itself varies from person to person, from circumstance to circumstance.

Scriptural examples of how physical evidence is not enough to understand spiritual Truth:

Exodus 16:26-30

26 Six days ye shall gather it; but on the seventh day, which is the sabbath, in it there shall be none.

27 ¶And it came to pass, that there went out some of the people on the seventh day for to gather, and they found none.

28 And the Lord said unto Moses, How long refuse ye to keep my commandments and my laws?

29 See, for that the Lord hath given you the sabbath, therefore he giveth you on the sixth day the bread of two days; abide ye every man in his place, let no man go out of his place on the seventh day.

30 So the people rested on the seventh day.

Excuse me, are these the people that God just showed mighty miracles, freed from slavery, and pretty much showed himself to? Yet some don't trust what he says...

1 Nephi 17:45

45 Ye are swift to do iniquity but slow to remember the Lord your God. Ye have seen an angel, and he spake unto you; yea, ye have heard his voice from time to time; and he hath spoken unto you in a still small voice, but ye were past feeling, that ye could not feel his words; wherefore, he has spoken unto you like unto the voice of thunder, which did cause the earth to shake as if it were to divide asunder.

Nephi's brothers had seen an angel, yet they continually doubted.

Related to this:

1 Nephi 15:7-11

7 And they said: Behold, we cannot understand the words which our father hath spoken concerning the natural branches of the olive tree, and also concerning the Gentiles.

8 And I said unto them: Have ye inquired of the Lord?

9 And they said unto me: We have not; for the Lord maketh no such thing known unto us.

10 Behold, I said unto them: How is it that ye do not keep the commandments of the Lord? How is it that ye will perish, because of the hardness of your hearts?

11 Do ye not remember the things which the Lord hath said?—If ye will not harden your hearts, and ask me in faith, believing that ye shall receive, with diligence in keeping my commandments, surely these things shall be made known unto you.

Heck, a good portion of the witnesses of the Book of Mormon, who saw the gold plates, fell away (some returning, some not, as far as I am aware). Without revoking their testimony concerning the plates, mind you.

As to how spiritual Truth is confirmed, there are a few scriptures I can cite, but I agree with Jake Toronto who commented that discerning what is spiritual communication and what not is essentially a life-long journey.

Doctrine and Covenants 8:2

2 Yea, behold, I will tell you in your mind and in your heart, by the Holy Ghost, which shall come upon you and which shall dwell in your heart.

Doctrine and Covenants 50:12-24

12 Now, when a man reasoneth he is understood of man, because he reasoneth as a man; even so will I, the Lord, reason with you that you may understand.

13 Wherefore, I the Lord ask you this question—unto what were ye ordained?

14 To preach my gospel by the Spirit, even the Comforter which was sent forth to teach the truth.

15 And then received ye spirits which ye could not understand, and received them to be of God; and in this are ye justified?

16 Behold ye shall answer this question yourselves; nevertheless, I will be merciful unto you; he that is weak among you hereafter shall be made strong.

17 Verily I say unto you, he that is ordained of me and sent forth to preach the word of truth by the Comforter, in the Spirit of truth, doth he preach it by the Spirit of truth or some other way?

18 And if it be by some other way it is not of God.

19 And again, he that receiveth the word of truth, doth he receive it by the Spirit of truth or some other way?

20 If it be some other way it is not of God.

21 Therefore, why is it that ye cannot understand and know, that he that receiveth the word by the Spirit of truth receiveth it as it is preached by the Spirit of truth?

22 Wherefore, he that preacheth and he that receiveth, understand one another, and both are edified and rejoice together.

23 And that which doth not edify is not of God, and is darkness.

24 That which is of God is light; and he that receiveth light, and continueth in God, receiveth more light; and that light groweth brighter and brighter until the perfect day.

Let me tell you that sometimes this understanding means knowledge, sometimes it means insight (I lack a proper translation of the german word I mean, I hope this one hits the intended meaning). Insight, for me, means an understanding that can't necessarily be communicated. For example, I gained a little insight in the power of God when contemplating just how huge and powerful the sun is, which God created. That was a powerful experience, but none that I feel can be properly communicated, because the result was mostly feeling, and is directly coupled to the experience.

So, I hope this shows that spiritual communication is not "just" feelings. But it is also not "just" knowledge, in the intellectual sense. (And really, how do you expect Truth to be confirmed, if not also through feelings? He doesn't want intellectually "convinced" people, he wants spiritually "converted" people) In the context Elder Ballard said what he said, I feel it was appropriate to word it like he did.

Elder Ballard is talking about the fruit of the Spirit - the "fruit" being a manifestation, consequence, or result of the presence of the Holy Ghost.

This teaching is founded on a few scriptures, mainly Galatians 5:22-23 which says:

22 But the fruit of the Spirit is love, joy, peace, longsuffering, gentleness, goodness, faith,

23 Meekness, temperance: against such there is no law.

and Ephesians 5:9 which says:

9 (For the fruit of the Spirit is in all goodness and righteousness and truth;)

Spiritual truths are confirmed by spiritual means. (I recommend perusing the answer at that link for an explanation.) Confirming feelings come by revelation from God and cannot be coerced or forced upon another to prove things to them.

However, when someone speaks what the Holy Ghost inspires (a testimony), the Spirit can manifest truth of it to those receiving it:

2 Nephi 33:1

... when a man speaketh by the power of the Holy Ghost the power of the Holy Ghost carrieth it unto the hearts of the children of men.

The Book of Mormon promises revelation to those who seek truth in several instances, for example:

2 Nephi 32:5

5 For behold, again I say unto you that if ye will enter in by the way, and receive the Holy Ghost, it will show unto you all things what ye should do.

In more modern times, Joseph Smith received a revelation about conducting church meetings:

D&C 46:7

7 But ye are commanded in all things to ask of God, who giveth liberally; and that which the Spirit testifies unto you even so I would that ye should do in all holiness of heart

The ultimate promise of revelation in the Book of Mormon is found in its closing verses as Moroni prepares to bury the record. It is probably the most-quoted passage by Latter-day Saints about how to know the truthfulness of the Book of Mormon:

Moroni 10

3 Behold, I would exhort you that when ye shall read these things, if it be wisdom in God that ye should read them, that ye would remember how merciful the Lord hath been unto the children of men, from the creation of Adam even down until the time that ye shall receive these things, and ponder it in your hearts.

4 And when ye shall receive these things, I would exhort you that ye would ask God, the Eternal Father, in the name of Christ, if these things are not true; and if ye shall ask with a sincere heart, with real intent, having faith in Christ, he will manifest the truth of it unto you, by the power of the Holy Ghost.

5 And by the power of the Holy Ghost ye may know the truth of all things.