How can "Mormons" say their church is the only true church?

Because of the restoration. A restoration is to bring something back. In the beginning God created Adam and Eve. Adam was the first prophet on the earth. A prophet is a man who speaks with God in behalf of the people on the earth. Throughout the history of the earth people have followed God and been led by a prophet and other times people have not followed God. Noah, Jacob or Israel, Moses, Isaiah, Elijah were some of the prophets in the time before Christ came to the earth. They instructed the people in their days in how to follow God's teachings. They prophesied that Christ would come to earth and atone for our sins. He did. Through Jesus Christ's teachings while on the earth, he taught us how to be happy, to keep the commandments, and to serve. He was an example of service, forgiveness, and compassion. He also established his church. He called 12 apostles. He gave these men the priesthood--the power to act in God's name. Then he atoned for our sins. He paid the price so that we can be resurrected and live with him again if we repent. After his death, Peter was the prophet. He had the priesthood authority, but he was killed and that authority was taken from the earth. The Great Apostasy followed when the people of the earth did not have a prophet. Many did continue to follow Jesus's teachings. Leaders of nations adopted Christian beliefs and created churches. Over time, great men sacrificed much including their lives so that all could have access to the Bible. There was a reformation. Men tried to decipher the best way to follow Christ's teachings. In America, religious freedom allowed all to choose how to worship. In the early 1800's in New York, Joseph Smith's family studied the Bible together. Members of his family attended different churches. Joseph Smith wondered which church he should join. He was reading in James 1:5, "If any of you lack wisdom, let him ask of God, that giveth to all men liberally, and upbraideth not; and it shall be given him." At the age of 14, Joseph decided to ask God. He went into the woods near his house and prayed. I don't know what kind of answer he thought he would receive. Two beings appeared to him: God the Father and His Son, Jesus Christ. He was told to join none of the churches, for none were correct and that in time he would help to restore the church that had been taken from the earth. Later the church was restored through Joseph Smith--the same church that Christ established when He was on earth. Joseph Smith was a prophet just like Adam, Noah, Isaiah and others. And we have a prophet today, President Thomas S. Monson. We have 12 apostles. We have pastors, teachers, evangelists. We have the priesthood authority as given to Adam and other prophets and as given to Peter.

I understand that you may not believe as I do--that the church has been restored through Joseph Smith and that The Church of Jesus Christ of Latter-day-Saints is Christ's restored church on the earth again. That is where the third member of the Godhead plays his part. The Holy Ghost is a witness of truth. Every person is invited to find out for himself/herself if what I have stated is true. I have prayed. I have studied. I continue to pray and study and ask. I have received a witness through the peaceful feelings of the Holy Ghost, through words in my mind, through a knowing within every party of me, that The Church of Jesus Christ of Latter-day-Saints is Christ's restored church on the earth. You can know too. And when I doubt, when I ask again, I am given an answer again.