Why did Mormons stop practicing polygamy?

The manifesto was only implemented initially to a limited extent. It was a statement by the church that the church would no longer practice plural marriage. However, plural marriages actually weren't under the jurisdiction of the church to start with. It wasn't until 1852 that this practice became general knowledge to the members of the church when Brigham Young and Orson Pratt introduced it at a special general conference. This was because all things in the church are to be performed by common consent. (D&C 26:2) It was unanimously accepted as a valid principle at that conference. So, just as the early priesthood leaders didn't need this to be something accepted by the church before 1852, they didn't have any need for it to continue in the church after 1890 either.

You could say it was a bit of a ruse to try and appease the government without actually being fully divested of it. Many priesthood leaders continued to utilize the priesthood organization and power to continue living according to the fullness of the New and Everlasting Covenant of Marriage, which of necessity includes the plurality of wives.

This needed to be preserved because the priesthood's ability to continue to qualify to receive the higher level of ministration of "thus saith the Lord" oracles from the Father and the Son was at stake. The level of ministration a priesthood or church body qualifies to receive depends upon what set of laws they are living in accordance to. (D&C 88:21-22; 76:77,86-88) If you do not keep the fullness of the Father's Celestial Law then you are unworthy of His presence and shall not receive of His ministration.

This is why prior to the Manifesto there was a separate priesthood body established ahead of time with a president and 6 councilors. This is based upon the organization called the School of the Prophets that was called to order as such in D&C 88:127-141 and D&C 90:7-15. This priesthood body was presided over by the person who was the Lord's Anointed who received the oracles, which are the "thus saith the Lord" level of revelations.

It is this organization of priesthood that is separate from the church that is at the root of the claims of Mormon Fundamentalists to have the authority to continue to perform plural marriages, among other things. Some called this organization the Council of Friends because those who enter into the School do so by the covenant of friendship and brotherhood as fellow "Sons of God". (D&C 88:133)

This organization created by John Taylor had a president and 6 councilors just as the earlier School of the Prophets did. It appears likely that this same organization was also called the Church of the Firstborn, but that needs further proof that I am not currently prepared to present. The important thing is to understand that there were bodies of authority separate from the Church of Jesus Christ of Latter-day Saints that were not subject to the church's authority.

One of the foremost factors that led to the Manifesto was the evidence that the vast majority of Mormons were personally opposed to plural marriage. This became clear when various versions of proposals for a state constitution for Utah were being drafted and voted upon. It was noted that plural marriage was a significantly negative feature to an approximate ratio of 13,000 opposed to 500 in favor. I'll dig out the cite for this later. I think it was mentioned in one of Lynn Bishop's books or Allred's book called "A Leaf in Review". It's been many years since I read those so my memory is a bit fuzzy.

Now, to address your question in relation to the warning given that those who do not abide the Covenant cannot enter into the Father's Glory: The evidence for this being the case is clear. Even when plural marriage was still being practiced during the mid 1800's there were other laws of the Father that were not being fully implemented either. The Saints struggled and tried to practice the Celestial economic laws but failed over and over. It was recognized that the Saints would have to be patient and wait for the due time of the Lord to bring about what scripture calls the redemption of Zion. (D&C 105:9-13)

So, the unfortunate reality is that we as a church are still under a level of condemnation. (D&C 85:54-58) This was affirmed as recent as 1986/1988 by Ezra Taft Benson who was the President of the Church at the time.

Benson, Ezra T Oct Conf. 1988, Nov ENSIGN, p. 6 par. 10:

Moses never entered the promised land. Joseph Smith never saw Zion redeemed. Some of us may not live long enough to see the day when the Book of Mormon floods the earth and when the Lord lifts His condemnation. (See D&C 84:54--58.) But, God willing, I intend to spend all my remaining days in that glorious effort. In the name of Jesus Christ, amen.

So, the truth is, we are yet awaiting the time that Zion shall be redeemed. We are in a state of damnation just as that passage warns. Celestial "thus saith the Lord" ministration has ceased and there won't be anymore received until the Saints finally come around to realizing the Father meant every single word He said. We are under condemnation because we have taken lightly that which we have been given. (D&C 90:3-5)

We must come to a point where we implement the full economic, social and educational systems that the Father instructed us to do per the "former commandments". If the Lord reveals light and truth and a people reject and turn away from it and take it lightly, then they are indeed under condemnation and shall eventually have taken from them all light and truth. (D&C 93:39; 60:2-3)

This is because they have one of two choices. They can hold and maintain integrity to what was given and press for repentance to that standard or they can turn to their pride and self-righteousness and twist everything into a mess trying to rationalize the avoidance of it. For example, they believe that breaking off a piece of the New and Everlasting Covenant is not actually breaking the New and Everlasting Covenant.

Unfortunately, this means that instead of looking forward to a period of great salvation and triumph when the calamities come, the Saints shall actually be unprepared and fully subject to the scourge and judgment that is coming if they do not repent in time. And, if we are slow to hearken to the words of the Father to heed this and other warnings, He shall be slow to hearken to our cries when we are driven into circumstances to humble us of our pride and self-righteous drunkenness so that at least some few of us can participate in Zion's redemption. (D&C 133:63-70)

What a perfect answer by Jason!

I would like to add more to what he has said about the Latter-day saints not currently living all God's commands and how that relates to polygamy.

In D&C 56:4 we have "Wherefore I, the Lord, command and revoke, as it seemeth me good; and all this to be answered upon the heads of the rebellious, saith the Lord."

In D&C 58:31-33: Who am I, saith the Lord, that have promised and have not fulfilled? I command and men obey not; I revoke and they receive not the blessing. Then they say in their hearts: This is not the work of the Lord, for his promises are not fulfilled. But wo unto such, for their reward lurketh beneath, and not from above.

It is clear from this that God gives commandments, like polygamy, to the Church in order that he may bless them. But when the church proves incapable of keeping the commandment the Lord, in order to reduce the level of condemnation the Church is under, revokes (takes back/removes) the commandment. The Church is therefore technically no longer under the condemnation of "breaking" the commandment although they will still suffer the consequences of not living the commandment.

A good example of how the Church is currently suffering as a result of not living this commandment is the number of unmarried sisters over the age of thirty (which is considered "old" in Mormonism). It has also led to many sisters marrying outside the Church as a compromise in order to not commit adultery.

So to address the term "Everlasting" it is clear that the Law of plural wives is really everlasting in that it is as needed today as the day the Lord revealed it to Joseph Smith. The only reason it hasn't actually "lasted" among Latter-day saints today is that the latter-day saints proved themselves unwilling to trust the Lord and keep His commandments as they did on many other occasions as described by Jason (to whose list I would like to add the policy on Blacks and the Priesthood as another example).

This is more of an addendum to Mason's answer, but is too long to fit in a comment.

I agree that the logical interpretation of "everlasting" is usually "something which persists forever", but that isn't always the case.

For example, we general consider the terms "eternal" and "endless" to be synonymous with "everlasting". The Doctrine and Covenants (part of the LDS canon) gives an alternate interpretation of these words that doesn't mean "unending" (D&C 19:10-12):

10 For, behold, the mystery of godliness, how great is it! For, behold, I am endless, and the punishment which is given from my hand is endless punishment, for Endless is my name. Wherefore—

11 Eternal punishment is God’s punishment.

12 Endless punishment is God’s punishment.

From this we can see that "Eternal" and "Endless" are variations of God's name. It's reasonable to extend this to include "Everlasting" as well.

With this in mind an "everlasting covenant" doesn't necessarily mean "a covenant which lasts forever", but rather "a covenant that is made with God".

The question of "why did Mormons stop practicing polygamy" is definitely an interesting one, but I just wanted to point out that an "everlasting covenant" can mean something other than "a covenant which never ends".