First, we may as well dispense with the semantic explanation: you do what feels the most true from your perspective, and I do likewise for myself. Of course we believe that what we do personally is the most 'true.' We would not have made our respective choices if we thought otherwise.
I will address the meaning of 'truth' only briefly. For the purpose of this post, I'll work on the assumption that there is absolute truth, that we are by nature designed to recognize right from wrong, that the majority of us desire to do right, and that those who choose (not merely to make mistakes, but conciously choose) to do wrong, must do so in rebellion to their own knowledge of these absolutes.
Truth is truth because it is absolute. While none of us will find every thread of truth in this brief lifetime, we're all born to recognize these threads when we see them, and by our nature we desire to collect and weave them into the fabric of our lives. This is what leads us to make the decisions we make. It especially guides our decisions concerning faith and religion-- the doctrines we choose to follow, the churches we join, our feelings toward opposing views. But ultimately, fundamentally, we're all doing the same thing. We're looking for the truth. Not one of us is more admirable than another, nor more favored of God, in that regard.
I admit that it has always bothered me when a member of my faith stands up and declares, "I know this church is true!" A church is an earthly organization structured to further the work and doctrine of God among His children. While its structure and purpose may be said to be divine, the organization itself is not the 'truth' for which missionaries give up years of their lives. The doctrine of Christ is true. His gospel is true. A church--his church-- is a vehicle to that end, not truth itself.
The Complete Church
That is the word that most accurately describes how I feel about the LDS faith, and it's what your Mormon friends most likely mean when they say "True Church."
There are innumerable things outside the LDS faith that are virtuous, lovely, of good report, and praiseworthy. Latter-Day Saints seek after these things too! We seek out the good wherever it may be found, in all people of all faiths and in every nation. Truth is everywhere, and there is no bit of truth that we do not seek. We do not believe that we own all the truth.
This is how the Church of Jesus Christ of Latter-Day Saints is organized. The rock-solid fundamentals of Christianity are its foundation. The window of heaven is open so God can direct modern matters through a living prophet (as the Lord has intended since the beginning of time). But nothing will be done currently that contradicts that rock-solid foundation; it can't, or the entire plan of God is frustrated.
I say emphatically, we do not believe that people of other faiths are wrong-headed, or fraudulent, or bad because they believe differently than we do. Quite the contrary-- our doctrine states that all mankind is included in this plan, from the most stalwart Christians to the generations of tribal islanders who lived and died having never heard the name of Jesus Christ. All good leads all of us to the same place, just not always at the same time, and God is fair.
There are no Mormons in heaven, nor Catholics, Protestants, Hindus, or Buddhists. Any who were deprived of the opportunity to receive Christ in this life will learn of him before the great day of Judgment. If you lived honorably, but despite your best efforts you did not find God and make your choice, He will find you and ask you Himself.
So the one great doctrine of the LDS Church is, in the words of Maya Angelou, "I did then what I knew how to do. Now that I know better, I do better." This is all God asks. Do what you know to do, and when you know better, do better. Do that, and there is no gift he will withhold from you, including salvation through Jesus Christ and every holy ordinance of heaven. Whether on this earth or out of it, God will find every honorable man and woman. He will find you and make the offer.
I know that the gospel of Jesus Christ, as taught from the foundation of this earth and in church on Sunday, is the truth. I know this because I've received a personal confirmation through the Holy Ghost of its truthfulness. I know it not because my church is exclusive, but because it isn't. I know it not because the house I'm living in is perfect, but because I'm working with a complete set of tools and building on a solid foundation, the rock of the Redeemer of Israel. What I do not know, I can learn. What I do not have, I can build. What I cannot do will be provided by the grace of God's Son.
As Mormons, we do not believe that we are products of a perfect church. We are only guardians, stewards, and grateful recipients of a complete set of instructions-- the same instructions which are available to every soul on earth who will receive them. Every soul, yours and mine equally.