Yes, this is true. Keep in mind that a hotel can't clean a room until they know it's vacant. If you've ever checked in early after a sold out night, it's probably due to the fact that the previous guest took the time to check out. You don't even need to go to the Front Desk. Just pick up the phone, dial "0", and tell whoever answers that you're leaving. It should only take 30-60 seconds. (If no one answers right away, just leave knowing that you tried to help out the next person staying in that room.)

I worked for years in hotels. Honestly, it doesn't really help the hotel out much. It really is just a simple way to possibly help out a stranger.