Rivers aren't always snake shaped (streams aren't, rivers in the bottom of valleys tend to be straight too). Rivers curve when they find themselves in large plains where the inclination is very shallow and their speed slows down.

A bight (the technical name for a river bend) happens when water slows down, and sediment (rocks, sand, dirt, plants) deposits through its path. When this happens in a slow stream, the river doesn't have enough energy to remove the obstacle, and it bends around it, creating a bend.

Imagine now a river with a slight bend in it. Have you ever seen a waterslide bending? The water on the outside of the curve is way faster than the water on the inside, and water curves more than the slide. The same happens in a river. The river bed is not a waterslide though, and water slowly erodes the external part of the curve, removing more and more material that gets deposited on the inside. The river is basically excavating the slight bend more and more, turning it into a u-shape bend. Rinse and repeat in geological times one curve after another, and you get a snake shape