As a little background, the Church of Jesus Christ of Latter-day Saints (LDS) is not the only "Mormon" church out there. There are other groups which believe in both the Book of Mormon, and in Joseph Smith the prophet. The largest of the other groups is most likely the Reorganized Church of Jesus Christ of LDS (RLDS). That was formed by a group of people who did not recognize Brigham Young as the legitimate successor to Joseph Smith as leader of the church, but rather believed the prophetic mantle was passed on to Smith's son, Joseph Smith III.

So, the question may be why should one believe that Young was Smith's legitimate successor?

First, to be perfected candid, my opinion is certainly influenced by the decision of my ancestors, who were "on the spot", so to say. My great-great-grandparents Colton joined the church around 1840, lived in Nauvoo, and were part of the pioneer trek west. If they had decided to follow Joseph Smith III, I may well be arguing from the opposite point of view. However, I think most of the people who were present at that time, were probably in a better position to judge between competing claims than we are now, and most people chose Brigham.

Second, I believe the scriptures (D&C) are clear that (a) the quorum of 12 apostles are equal in authority to the First Presidency, and that (b) when the First Presidency is dissolved (as in by the death of the president), the 12 apostles should take the leadership role in the church. Brigham Young led the church in his capacity of President of the Quorom of the Twelve for many years before being sustained as the new President of the Church.

Third, I believe in the miraculous event that took place after Joseph's death as Sidney Rigdon and Brigham Young were debating who should lead the church, known as the "Transfiguration of Brigham Young." Basically what happened was that as Brigham was speaking, his face and voice became those of Joseph Smith for a few moments.

Here is a first hand account, written by B.F. Johnson:

(from http://www.ldstemplepage.org/bfjohnso.html)

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And I do further bear as a testimony, faithful and true, to the Church and to all the world, that at a conference of the whole Church, at Nauvoo, subsequent to the Prophet's death and return of the absent Apostles, that I sat in the assembly near to President Rigdon, closely attentive to his appeal to the conference to recognize and sustain his claim as "Guardian for the Church." And I was perhaps, to a degree, forgetful of what I knew to be the rights and duties of the apostleship, and as he closed his address and sat down, my back was partly turned to the seat occupied by Apostle Brigham Young and other Apostles, when suddenly, and as from Heaven, I heard the voice of the Prophet Joseph, that thrilled my whole being, and quickly turning around I saw in the transfiguration of Brigham Young, the tall, straight and portly form of the Prophet Joseph Smith, clothed in a sheen of light, covering him to his feet; and I heard the real and perfect voice of the Prophet, even to the whistle, as in years past caused by the loss of a tooth said to have been broken out by the mob at Hyrum. This view, or vision, although but for seconds, was to me as vivid and real as the glare of lightning or the voice of thunder from the heavens, and so deeply was I impressed with what I saw and heard in this transfiguration, that for years I dare not publicly tell what was given me of the Lord to see. But when in later years I did publicly bear this testimony, I found that others would testify to having seen and heard the same. But to what proportion of the congregation who were present I could never know. But I do know that this, my testimony is true.

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Fourth, I also believe that the subsequent growth of the LDS church as compared to the other Mormon denominations, is a testament to this church being the "real deal". LDS Mormons (most anyway) believe that the church is a fulfillment of Daniel's prophecy of the stone cut out without hands, which grew to fill the whole earth. Thus, the growth of the church to "fill the whole earth" has been prophesied. I don't know how the other Mormon denominations view the Daniel prophecy, if they think their church will also fill the whole earth..

Fifth, I believe I have had a spritual witness that this church is the "authorized version".

There's probably a few other reasons, but I'll leave my list at that.