Unpacking the question:
This question has been coming up lately from two groups who might be surprised how much they
have in common with each other. Religious conservatives often bring up the belief to show that Mormons are unorthodox and blasphemous and shouldn't be listened to. Secular cultural liberals don't see "unorthodox" or "blasphemous" as negative labels, so they bring up the belief to cast Mormons as weird (secular for "unorthodox") and arrogant (secular for "blasphemous") and not worth listening to (no translation necessary).
Now, someone who is trying to say Latter-day Saints aren't worth listening to is probably not going to listen to an answer to this question. Which is too bad, because it's actually a question that gets at some of our most fundamental beliefs.
And while I will freely admit that this particular LDS belief is far outside the cultural mainstream, I think it is a greater influence for good on us than most observers realize.
Most Western religions teach that we are primarily God's creations: beings he made (in his image, no less!) to find joy and to help complete the beauty of the universe. Most Eastern religions teach that we are actually a part of God: that a portion of him is manifest in every being and that we will eventually merge back into God as a drop returns to the sea.
Latter-day Saints join Western religions in treating humans as beings who exist separate from God, but join Eastern religions in interpreting our true natures as radically divine. In our view, the soul is the seed that contains the tree. Or the tiny atom whose mass quietly contains incredible quantities of energy. God, as a distinct and separate individual, is with us, watching over us, and communicating with us--but his full Divinity is also in us, waiting to be allowed to unfold.
This view is not an obscure Mormon teaching--it's a basic assumption for many Latter-day Saints. And it informs the way we look at several issues. For example:
Parenting. One reason family matters so much to Latter-day Saints is that we believe so fully in parent-child relationships as a manifestation of the divine. We believe our souls are still children even as our bodies become parents, but that in earthly parenting we can experience a portion of God's identity and unlock our own divinity in the process. Learning to be a good parent is, in Mormon terms, the most direct route toward learning to be like God.
Facing Adversity. For most Western religions, God created the universe. For most Eastern religions, God is the universe. And either of those views raises a significant question: if God is good, why do bad things happen? In a Latter-day Saint view, the cores of our souls (called "intelligences") are eternal in both directions: they have always existed and will always exist. God didn't create himself and he didn't create the innermost core of us--he is a God partly because he so fully understands us and processes by which we can grow into our divinity. And so for Latter-day Saints, adversity is not a flaw in the universe so much as an opportunity for growth. Our reputation for rising to face difficulty comes in part from the example of our history, but also from our beliefs about the nature of our own souls.
Sin. There are numerous reasons to believe sin is bad. There may be consequences. It's against the rules. It will disappoint someone you care about. It can hurt people. But our belief in the endless potential within every human being allows us to focus on another reason sin is bad: sin damages us, it dams our eternal progression. And so after we sin, we don't just seek forgiveness--we are looking for healing, asking Christ to help mend and nourish our damaged divinity. And when we live good lives, we can feel how much more they're in harmony with the impulse of growth within us.
So do Mormons believe we can become Gods in the afterlife? Absolutely. But it's not like we sit around all day and dream about that future in cartoon terms. For most Latter-day Saints, our divine potential matters now: as we watch our children grow, when we face trials and adversity, when we seek healing through Jesus Christ's Atonement.
And if anything, I wish we would talk about this controversial part of our doctrine a little bit more. One thing I think Latter-day Saints have to offer to our neighbors and friends of other faiths is just a little sense of wonder about whether our most sacred humans traits--our interest in creating and organizing, our capacity for empathy and imagination, our longing to nurture and to serve, our innate abhorrence for cruelty and evil--are really signs of a radical godliness that is already part of every human soul.