Natural selection is a mechanism that can be proven in a very simple argument.
We know that mutations happen, because they cause genetic diseases such as cystic fibrosis in humans, or hip dysplasia in dogs. We also know that the rate at which such mutations occur is not slowing down appreciably. We do know, however, that genetic diseases cause much suffering, often make it difficult to lead a normal life, and are fatal in some cases. Given that we know that genetic mutations keep occuring, and that they inherit faithfully when rare, we must ask ourselves why we haven’t all died from genetic disease yet. The only possible answer is that those individuals who carry genetic diseases are not able to reproduce to quite the same extent as healthy individuals. And that is natural selection.
This can also be phrased as an argument from contradiction for those who like mathematical proofs. It basically goes like this: assuming that natural selection is absent, and genetic mutations keep occurring, the human species should have vanished soon after its emergence; ditto for all other species. Yet, after 200,000 (for humans) to 3.5 billion years (for life itself, and little less for many ancient species), we are still here. Which is a contradiction.
Favourite piece of trivia: Did you know that “evolution” written backwards spells noitulove?