Why do some people actually click A to auto?

A clicking is something that came from RTS games in order to do two things: a) avoid misclicking b) to attack-move and kite well. If you attack-move/orb walk by cancelling the AA animation perfectly, you can cover the most distance between each unit (or in this case, champion) while not stopping to attack.

I personally prefer to A-click to attack as I came from Starcraft/Warcraft/Dota and naturally carried it to League.

because sometimes when you right click something you will walk to it instead of attacking it (more noticeable on range champs). The a left clicking prevents that

Fastest action of attacking. If you press A and walk into a bush, it will instantly auto attack any enemy within your range/vision.