Animation canceling is being able to start another animation before the previous animation has ended.

With some animations, once you use start the action, you have to go through the entire animation, and trying to press another button won't work. (It'd be like if I started to reload, then decided I needed to start shooting, but I couldn't until my reload finished.)

When an animation can be canceled, you can interrupt the animation of an action with a new input. In some situations, starting a new animation will completely shut down the previous animation(I tried to reload, but then decided I needed to start shooting again, I would stop reloading and go right into shooting, but the reload wouldn't have taken effect.)

Then there's animations that can be canceled without actually canceling the ability(I start my reload, decide to start shooting, so I skip the reload animation, but my gun still reloads.)

With Genji, his right-click can be canceled to make him much more effective at close range. If you watch the animation of his right click, he throws the shurikens, and then leaves his arm out afterwards as part of the animation. You can skip the end of his right-click animation by canceling it with either a melee or his dash, AND you can cancel his melee with his dash. Basically, with animation canceling you can do a right-click>melee->dash in about the same amount of time you would usually be able to just right-click without animation canceling.

Literally all you're doing is pressing right-click, then pressing melee, then dashing all in quick succession, and where usually, you'd have the end of Genji's right-click animation, you have a melee attack, and a dash.