According to this Irelia guide, if you cancel the animation early then your next auto-attack will start sooner (I could be interpreting it wrong - it could simply mean that staying close to your opponent will allow you to do more damage).
Basically put, the auto attack animation will apply damage before it is fully finished, so you can cancel it halfway through, move up a little, and reactivate the auto attack. It is difficult to master, but once you do learn it, you can attack 3 times in what would normally be 2.
My question is, can you effectively increase your attack speed with AA animation cancelling? If you time the animation cancel right by moving in between autos, can you attack faster? As a dedicated AD Carry player, I feel like I should know this by now.
Note: For info on what animation cancelling is, see this page.