In Red Dead Redemption, during the poker game, you can look around at other people's faces. I assume there's a use of this option, but I wasn't able to get any hint when it may be. Does it actually have a purpose, or is it just for game immersion ?
Well technically it has a purpose and also adds to game immersion ;)
In all seriousness the NPC players give off tells as well as change their tones/mannerisms slightly depending on past hands and their current hands. The look around option gives you a chance to watch for those tells and play accordingly. It can also give you a better idea of when to cheat, since most of the time the other players will catch you (if you're as bad at it as I am).
If you look around after several hands you can tell who is bluffing and who has a good hand. You can also use it to determine if you can get away with cheating.