Please note that while the animation gets reduced, there is no increase in DPS. League of Legends uses a Internal Cooldown on Autoattacks or as Riot calls it Attack Delay Offset Percent.
This is not DotA, the animation and the ICD are completely independent.
If you want proof, stutter step while attacking inhibitors. The way they are programmed is that if you click on an inhibitor with your champion while your ICD is still cooling down, you will continue moving until the ICD ends and then they will autoattack. This is unlike the behaviour of everything else in the game where as soon as you right click your champion stops and then waits for their ICD and then attacks.
Do not spread misinformation that this is a DPS increase, it is not. It's been tested.
And the reason you want to do this is threefold.
1) it helps you reposition if you need to either escape or get closer to an enemy while still dealing the same amount of damage 2) it helps you be more unpredictable 3) it's good to keep your APM up because you should always be stutter stepping at every point in the game, there is actually no reason not to. As long as you are still accurate, stutter stepping has zero downsides but with many up sides.
Gamer Logic - Anyone worse is a noob, anyone better is a tryhard /v/irgin youtube.com/user/Shinius