There are a lot of myths both inside and outside Mormondom about the LDS claim to be the “one true Church.” That claim is difficult for people of other faiths to accept for obvious reasons. But some Mormons may struggle with it as well because it may seem difficult to reconcile that claim with our belief in a just and merciful God who loves all his children equally.
I do not expect it will ever be “easy” to accept the “one true Church” claim. However, I think Church members can unfortunately make it more difficult to accept the “one true Church” claim than is necessary because they jump to some unnecessary conclusions about what the “one true Church” claim actually means.
In order to correctly understand what the “one true Church” claim truly means, we first need to be clear about what that claim does not mean. Although the LDS Church unapologetically claims to be the exclusive holder of priesthood keys, Mormons do not claim any monopoly on truth or communication with God. To the contrary, Mormons believe that God calls and guides leaders of all faiths to deliver truth to his children and accomplish his great work, and that God loves and guides all who sincerely seek him, regardless of what religion or church they belong to. The following is my attempt to bust some of the many myths about what the “one true Church” claim means by using the words of Church leaders and Church publications.
Myth: God stopped communicating with mankind for hundreds of years during “the Apostasy,” when the priesthood was not found upon the earth.
“The line of priesthood authority was broken. But mankind was not left in total darkness or completely without revelation or inspiration. The idea that with the Crucifixion of Christ the heavens were closed and that they opened in the First Vision is not true. The Light of Christ would be everywhere present to attend the children of God; the Holy Ghost would visit seeking souls. The prayers of the righteous would not go unanswered.” Elder Boyd K. Packer 
“All down the ages . . . good and great men, not bearing the Priesthood, but possessing profundity of thought, great wisdom, and a desire to uplift their fellows, have been sent by the Almighty into many nations, to give them, not the fulness of the Gospel, but that portion of truth that they were able to receive and wisely use.” Elder Orson F. Whitney, quoted by Elder Howard W. Hunter 
Myth: In Joseph Smith’s day, the Christian churches were completely corrupt and were an abomination to the Lord.
“Informed Latter-day Saints do not argue that historic Christianity lost all truth or became completely corrupt. The orthodox churches may have lost the ‘fullness’ of the gospel, but they did not lose all of it nor even most of it. . . . [T]he actual LDS view,  is that the orthodox churches are incomplete rather than corrupt. It is their postbiblical creeds that are identified in Joseph Smith’s first vision as an ‘abomination,’ but certainly not their individual members or their members’ biblical beliefs.” Stephen E. Robinson (as quoted on www.lds.org)
Myth: Because Mormons are the only people currently guided by true prophets with the priesthood authority, they are the only people eligible to receive God’s inspiration.
“All men share an inheritance of divine light. God operates among his children in all nations, and those who seek God are entitled to further light and knowledge, regardless of their race, nationality, or cultural traditions.” Elder Howard W. Hunter
Myth: As God’s covenant people, Mormons are the people God has chosen to perform his great and marvelous work.
“God is using more than one people for the accomplishment of His great and marvelous work. The Latter-day Saints cannot do it all. It is too vast, too arduous for any one people. … We have no quarrel with the Gentiles. They are our partners in a certain sense.” Elder Orson F. Whitney, quoted by Elder Ezra Taft Benson
Myth: Although well-intentioned, leaders of other religions and churches are unwitting servants of the devil who are deceiving and leading God’s children astray.
“We believe that most religious leaders and followers are sincere believers who love God and understand and serve him to the best of their abilities. We are indebted to the men and women who kept the light of faith and learning alive through the centuries to the present day. . . . We honor them as servants of God.” Elder Dallin H. Oaks
“The great religious leaders of the world such as Mohammed, Confucius, and the Reformers, as well as philosophers including Socrates, Plato, and others, received a portion of God’s light. Moral truths were given to them by God to enlighten whole nations and to bring a higher level of understanding to individuals. … We believe that God has given and will give to all peoples sufficient knowledge to help them on their way to eternal salvation.” Elder James E. Faust
Myth: People belong to either the LDS church or the “Church of the Devil,” also known as the “great and abominable Church.”
“[I]ndividual orientation to the Church of the Lamb or to the great and abominable church is not by membership but by loyalty. Just as there Latter-day Saints who belong to the great and abominable church because of their loyalty to Satan and his life-style, so there are members of other churches who belong to the Lamb because of their loyalty to him and his life-style. Membership is based more on who has your heart than on who has your records.” Stephen E. Robinson (Ensign magazine)
“We must understand . . . that not everyone is going to accept our doctrine of the Restoration of the gospel of Jesus Christ. For the most part, our neighbors not of our faith are good, honorable people-every bit as good and honorable as we strive to be. They care about their families, just like we do. They want to make the world a better place, just like we do. They are kind and loving and generous and faithful, just like we seek to be.” Elder M. Russell Ballard
Myth: If a person doesn’t join the Church after learning about it, that person must not be honest in heart.
“Perhaps the Lord needs  men on the outside of His Church to help it along. They . . . can do more good for the cause where the Lord has placed them, than anywhere else. … Hence, some are drawn into the fold and receive a testimony of the truth; while others remain unconverted . . . the beauties and glories of the gospel being veiled temporarily from their view, for a wise purpose.” Elder Orson F. Whitney, quoted by Elder Ezra Taft Benson
 Boyd K. Packer, “The Light of Christ,” Ensign, Apr. 2005, 11 (quoted on Church website at: http://www.lds.org/ldsnewsroom/).  Orson F. Whitney, Conference Report, Apr. 1921, pp. 32-33 [quoted by Howard W. Hunter, “The Gospel-A Global Faith,” Ensign, Nov 1991, 18].  Craig L. Blomberg and Stephen E. Robinson, How Wide the Divide? A Mormon and an Evangelical in Conversation (Downers Grove, IL: InterVarsity Press, 1997), 61 (quoted on Church website at http://www.lds.org/ldsnewsroom/).  Elder James E. Faust, “Communion with the Holy Spirit,” Ensign, May 1980, 12 (emphasis added).  Howard W. Hunter, “The Gospel-A Global Faith,” Ensign, Nov 1991, 18 (emphasis added).  Alma 29:8 (emphasis added).  Orson F. Whitney, Conference Report, April 1928, p. 59 [quoted by Ezra Taft Benson, “Civic Standards for the Faithful Saints,” Ensign, Jul 1972, 59].  Dallin H. Oaks, “Apostasy and Restoration,” Ensign, May 1995, 84.  Elder James E. Faust, “Communion with the Holy Spirit,” Ensign, May 1980, 12 (emphasis added).  Stephen E. Robinson, “Warring against the Saints of God,” Ensign, Jan 1988, 34.  M. Russell Ballard, “Doctrine of Inclusion,” Ensign, Nov 2001, 35.  Orson F. Whitney, Conference Report, April 1928, p. 59 [quoted by Ezra Taft Benson, “Civic Standards for the Faithful Saints,” Ensign, Jul 1972, 59].