Based on my experience as a gamer, clutch is handling difficult situations. Let's say... you play a game of paintball. Your friends lost, and you're left alone to finish off 5 more players. With a skillful play, you manage to take out all 5 of the other players and win the round.
To sum it all, clutch is: Winning the round when the odds are stacked against you. The more the odds are stacked against the player or the more important the round win, the bigger the clutch.