Can champions increase attack speed by auto-attack cancelling?

Solution 1:

Your entire attack animation consists of three different kinds of frames: your initial swing, the hit, and the backswing. Animation cancelling consists of cancelling the backswing portion of the animation, usually by moving, to get in to a better position to continue attacking.

Keep in mind though, even though you cancel the backswing, there's still a built in-cooldown before you can attack again that's identical whether or not you choose to cancel. The main advantage, as well as the point trying to be made in the Irelia guide, is that by cancelling it allows you to reposition more aggressively(move forward) so that if a champion was retreating that can result in additional attacks.

Certain champs can cancel in to skills to effectively "double" their auto-attacks though since it resets their auto-attack cooldown, such as Sivirs bouncing blades, or Jax's Empower. Effective use of this is very important to maximize damage.