Lamarks theory basically said that to acquire a trait in an animal that would be used to benefit it's survival, the trait would gradually grow over time. Let's take the classic giraffe example. You have a giraffe with a shot neck and it needs to reach the trees. Over time, the more it reaches the more it's neck grows and that trait is passed down. That was Lamarks theory. Darwins theory however said that the giraffes with the longer necks to begin with would live and eat and the giraffes with the shorter necks would die off and eventually over time there would only be the long neck giraffes and that would better their survival. As for evidence pointing to natural selection, there's a lot of it, but we'll go with drawins findings of the finches in the Galapagos. On different islands with different climates, the finches beaks and other body parts adapted so they could find and eat food easier. This was evidence of natural selection.
Sorry for such a long answer. I hope you ca get something useful out of it!