Mormon Doctrine about the Fall of Adam
Mormons believe that the Fall of Adam was a necessary event in the history of this earth. When Heavenly Father placed Adam and Eve in the Garden of Eden, he commanded them to multiply and replenish the earth. He also commanded them not to eat of the fruit of the tree of knowledge of good and evil, but gave them their agency if they wanted to choose to eat of that tree's fruit. He told them that if they did eat of the fruit of the tree of knowledge of good and evil, they would "die."
The word "die" sounds like a very severe punishment. Indeed, most of the Christian world believes that eating of that fruit was a sinful act and therefore merited a severe punishment to Adam and Eve and their posterity. Mormonism teaches that Eve ate of the fruit by being deceived by Lucifer, because she wanted more knowledge and understanding of her purpose on earth than she enjoyed at that time. So she was willing to take the risk of using her agency to disobey that commandment, because she believed (rightly so) that there must be more to existence than living in a garden forever.
Adam was not deceived by Lucifer, but he was aware that he had been commanded to stay with Eve and to have children. When Eve told him that she had eaten some of the "forbidden fruit," he knew that he should also eat some of it, so that they could stay together and complete the purpose they had been given of multiplying (having children) to populate the earth and fulfill Heavenly Father's plan.
Mormons believe, through the teachings of the Book of Mormon, that Adam and Eve needed to have changes in their bodies in order to have children (See 2 Nephi 2:21,22). Those changes could only happen after their bodies became subject to death. The Fall of Adam brought the consequence of death into the world, as a necessary step in the plan of salvation.
Soon after Adam and Eve were cast out of the Garden of Eden, they were taught about the atonement of Jesus Christ, and about the resurrection of all mankind that would be brought about through the death and resurrection of Jesus Christ. They understood that they had made a good choice in eating of the forbidden fruit, because it allowed them to have children and to begin progressing as mortal beings who would be tested while living away from the presence of God.
Mormons believe that Heavenly Father adheres so very strongly to the use of personal agency (right of free choice), that he gave Adam and Eve what seemed to be contradictory commandments. The reason is, simply, that he wanted them to have a choice as to whether they would accept a plan that included their eventual death on this earth, rather than staying forever in the garden of Eden. He did not want to make that choice for them.
Mormons discuss the eating of the forbidden fruit as a "transgression," not as a "sin." It was not a sinful act, but a necessary act which Eve and then Adam bravely took upon themselves. Many centuries later, Moses described a vision where he saw Adam and Eve rejoicing after they had partaken of the forbidden fruit, because they had begun to understand "the joy of our redemption, and the eternal life which God giveth unto all the obedient." (See Moses 5:10,11)
A Book of Mormon prophet taught his children, "Adam fell that men might be; and men are, that they might have joy." (2 Nephi 2:25)