If you search YouTube for "Ciderhelm attack move" you'll find a very good video that explains it all really well.
Basically the ranged autoattack cycle goes like this:
- Play pre-AA animation
- Fire projectile (the AA timer starts now)
- Play post-AA animation
- Wait for AA timer to finish.
If you only right-click your target, your champion will not move forward during step 4 unless the target leaves your AA range. Try this in a custom sometime. As Caitlyn, walk right up next to the Scuttler Crab and start attacking. Cait won't chase after it until it runs out of AA range. If you immediately click forwards the instant you've fired the bullet, you'll spend step 3 and step 4 moving instead of standing still.
Ideally you will click your target again as soon as your AA timer is up, but even if you don't, it's still worth it. You'll be moving forward when you could be autoing, but you're keeping up with your target while you do so.
It's even more important if you are not directly behind your opponent. Imagine that you're purple side and they're at the top edge of the (warded) bottom bush of bot lane. You're standing in the center of the lane. Your support engages and you know you win this fight, so you want to keep hitting them as they run southwest to their turret. You don't want your champion to run towards them; you want to run parallel to them down the length of the lane so that they remain in range for longer.