What is Animation Canceling?

Put simply, it's buffering your commands so that you can get your abilities/auto attacks faster.

Examples of animation Canceling

Kalista's rend has an animation, however if you cast it while using her passive the animation is canceled and it still goes through.

Xayha's e can be cast while throwing her q out to make her feathers come at separate times and it's a very hard to dodge snare.

Sivir's w can be used immediately after an auto attack is fired so she gets two autos right after each other.

Why animation Cancel?

Efficiency. If you get good at this you can up your damage output significantly. (It also makes you look cool :P)