How do LDS know that Brigham Young and his successors are true prophets?

Short answer: They trust that God is in control and is ultimately responsible for ensuring that His prophet is in charge of the restored One True Church.

From http://www.lds.org/prophets-and-apostles/unto-all-the-world/prophets-of-the-restoration?lang=eng:

Since Heavenly Father and Jesus Christ ushered in the Restoration of the gospel and re-established the Church, the voice of the President of the Church has been a witness of divine, eternal truth.

Several LDS Scriptures support the idea as well, and are all listed at http://www.lds.org/scriptures/gs/prophet?lang=eng

A small sample:

• If there be a prophet, I the Lord will make myself known unto him in a vision, Num. 12:6
• I ordained thee a prophet unto the nations, Jer. 1:5, 7
• God hath set prophets in the Church, 1 Cor. 12:28 (Eph. 4:11).

In addition to the above, it may be that the definition of Prophet that you and I understand is not the same definition given in LDS theology. So, for the record, here it is:

A person who has been called by and speaks for God. As a messenger of God, a prophet receives commandments, prophecies, and revelations from God. His responsibility is to make known God’s will and true character to mankind and to show the meaning of his dealings with them. A prophet denounces sin and foretells its consequences. He is a preacher of righteousness. On occasion, prophets may be inspired to foretell the future for the benefit of mankind. His primary responsibility, however, is to bear witness of Christ. The President of The Church of Jesus Christ of Latter-day Saints is God’s prophet on earth today. Members of the First Presidency and the Twelve Apostles are sustained as prophets, seers, and revelators.

There is a rather important scripture from the Old Testament that @DavidStratton missed.

Amos 3:7 (KJV): Surly the Lord God will do nothing but He revealeth His secret unto His servants the prophets.

As the LDS church see's it, if the true chruch of Christ is on the earth, there Absolutely must be a prophet on the earth.

Now to answer your question: In the Doctrine and Covenants, the Lord laid out the foundation for how the church leadership was to be continued.

The person who has been in the Quorum of the Twelve Apostles the longest, is at the death of the old prophet sustained, and ordained the new prophet of the church. If you look at the history of the Quorum of the Twelve during Joseph Smiths day, Brigham Young was the senior apostle.

So for the general LDS population, everyone is instructed that they should come to their own belief in not just Joseph Smith but also the current prophet of the church. However if you have a testimony that Joseph Smith was a prophet and the revelation he received was from God, then it follows that as long as that revelation was followed each consecutive president would also be a prophet of God.

Little historical notes:

The history is a little messy because of the explosive growth of the church and the many people who either left/were excommunicated during that time. But he (Brigham Young) was the longest serving apostle and the president of the Quorum when Joseph passed away.

As a note of interest Hyrum would actually have been the next prophet if he had not died with his brother Joseph. However he was not the president of the Quorum of the Twelve because he was the 1st counselor in the 1st presidency of the church. And being the president of the Quorum of the Twelve even today is not necessarily an indication that you are the longest served apostle.

Interesting that no one has mentioned that the only person Joseph Smith favored to succeed him was his brother, Samuel, who died shortly after the martyrdom--probably murdered by Hosea Stout by poison, in an attempt to delay succession to the presidency until Brigham Young returned to Nauvoo from Boston. Both Samuel's daughter and brother William believed that Stout was acting under orders from Young and Willard Richards.

The enmity between Emma Smith and Brigham Young is well known, and Emma never recognized Young as the rightful successor to her husband. The real Emma is far from the Emma the Church portrays today, as is Brigham Young. I doubt many Latter-day Saints could gain much of a testimony is they were asked outright if they sustained Young as a true prophet of the Lord, or have even read the Journal of Discourses, and studied his heretical doctrines, such as of blood atonement.

Current LDS claims of succession are not based on any historical evidence, but only on the belief that the Twelve and not the Nauvoo High Council comprised the leadership of the Church. Even then, Young was not sustained as president and prophet until 1847, more than three years after the martyrdom. In other words, the LDS pioneers saw no need for succession or a prophet until they reached Utah.