When in a certain situation, organisms with traits more suited to survive in that situation often survive more often and are able to reproduce more, which means that their genes eventually dominate the gene pool.
For example: take a light moth population and a dark moth population. They live in a forest with dark trees, so the dark moths blend in more and are harder to find and hunt. As a result, more of them survive than the light moths, so more of them are reproducing and passing their genes along.
Over time, those genes may become the only traits of that variety that the organism can have, so they grow to evolve and adapt to their surroundings.