Why does a convex lens converge light and a concave lens divert light?

The first source is a visual that shows how the light bends. When light goes from a lighter medium to a denser medium it bends bends because it slows down. Where it hit first slows down first and where it hits next slows down next.

Picture a long chain stretched across a frictionless plane. It is straight across a path moving down the path. When you slow down the first link first, the next link will slow down next and so on. As the links slow one by one the chain changes direction and heads at an angle towards the first part that slows down.

Converging lenses change the direction towards the middle. This makes the arrangement of light the same but in a closer area. Out eyes perceive this to be a larger image.

Diverging lenses do the opposite. This gives the human eye a perception of things being smaller.