What happens to Finny and how?

This really depends on when in the story you mean. The major event is when Gene and Finny do their "Super Suicide Society of the Summer Session". Finny falls out of a tree after Gene shakes a branch on purpose. THe result is Finny becoming permanently disabled. Finny ends up taking another fall later in the book and subsequently dies.

Finny is an athlete, one of the best. He doesn't like rules and is very impulsive, something that tends to get him into a bit of trouble. It's this impulsiveness that leads to Finny's accident and the shattering of his leg. One poor decision leaves him unable to ever again participate in the sports he loves.

When Finny falls from the tree it's an antic that he and Gene were supposed to do together, but Gene jostles the tree and Finny falls. Gene's sense of guilt takes over for the remainder of the novel, and Finny's greatest wish is that Gene take over for him. They become as one person. Finny using Gene to accomplish the things he never will, and Gene spending years trying to make up for the things he feels responsible for.