It's a big, transcendent deal for Latter-Day Saints too, just in a different way. Mormons believe that God the Son, Jesus Christ, has a physical body, but before the Incarnation he didn't.
As I understand it, the Catholic doctrine of "hypostatic union" teaches that Christ was both fully human and fully divine during his mortal ministry. LDS doctrine essentially takes that idea one step further: while Jesus has always been the Son of God, participated in the Creation alongside the Father, and gave revelations and power to his prophets and his people in ancient days, he has also always been a son of God, as we all are, and followed the same eternal path of progression as we do, serving not only as our Savior and our Lord, but as our Exemplar.
He began as a spirit child of God, as we all did, was born into mortality, lived among us and experienced the joys and hardships of mortal existence, and eventually died, just like the rest of us. But then he did something that only he, being God, was capable of: he broke the bands of death, taking back his life and opening the doors of the Resurrection, so that all of us can follow in his footsteps and continue our progress.
Latter-Day Saints believe that the righteous can achieve "exaltation," the quality of eternal life that God the Father and Jesus Christ enjoy, in the hereafter, but saying that it will be in "our fleshy bodies" is a bit of a simplification.
When Jesus was resurrected, he appeared to the apostles in his resurrected body. He showed them the wounds from the crucifixion and had them touch and handle him so they would know he was not some sort of ghost, and even ate food with them. (See Luke 24.) But he said something very unique during this meeting: "a spirit hath not flesh and bones, as you see me have." The much more common description of our physical bodies is "flesh and blood;" by saying "flesh and bones" instead, this implies that, while our resurrected bodies will still be distinctly our own bodies, (note the crucifixion wounds,) they will be different in quality in some way. Mormons believe that these bodies will be perfected and immortal, incapable of further corruption or death.
This shows what the profound implications of the Incarnation are to Latter-Day Saints: when Jesus said "I am the way, the truth and the life, and no man comes to the Father but by me," he was describing himself as our Exemplar. He showed us the way to "come unto the Father" and achieve exaltation by achieving it himself!