Going along with what Toast said, Attack-move (a-click) is a mechanic popularized by Starcraft and Warcraft. A-move would allow units to automatically attack all enemies along the way.
The main advantage to A-moving is that you don't have to move your mouse. The milliseconds it takes to recognize a target and physically move the mouse above it makes a noticeable difference in professional play. Just let the keyboard hand and computer handle it.
- My champion is in the lower-left corner of the screen
- Enemy champion is in the middle of the screen
- My mouse is in the top-right corner of the screen
And my champion, the enemy champion and my mouse make a straight line. So now I have two choices,
Right-click on the enemy: I need to move my mouse down to the middle of the screen and click within the hit-box of the enemy, about 50 square pixels out of the whole screen. If I miss the hit-box slightly, my champion walks helplessly towards the enemy.
Attack-move: I hit "A" on the keyboard and then I left click where my cursor already was. I didn't have to drag my mouse anywhere. I didn't have to aim my cursor. I saved several milliseconds.
A-moving is especially useful when you are kiting because, by shaving off those milliseconds of mouse movement, you can use a champions attack speed to its full potential while moving.
It's useful in general when you have a good position on your enemy and don't have to retarget it every time,
It's similar to why you would use QWER to cast spells instead of clicking the icons each time.
Doublelift and many other pros may have previously played DOTA, Warcraft or Starcraft where APM optimization is the norm and developed a habit of A-moving.