The term "clutch" originally comes from the broadcasting of sporting events and describes a player who keeps a cool head and performs well under very high pressure (even when the whole match or tournament is on the line). The best example would probably be the decisive penalty kick in a World Cup final.
The meaning of clutch has been adopted 1 to 1 in e-sports and is now particularly common in turn-based games such as CS:GO, where there can be situations with large outnumbers or game-deciding match-ups of individual players.
In common usage, the word clutch in gaming is most often used for a round that a player decides for himself and his team, even though he had to play against the entire opposing team. However, it can also be used for a player who delivers extremely in the last crucial minutes of a match, helping his team to an often surprising victory.
Any situation in a multiplayer game has the potential for "clutch plays" when a lot is at stake and the performance of a single player can have a strong impact on the entire game. In most games, winning such situations is called clutch.
The term "clutch or kick" means that teammates will start a vote to kick you out of the game if you don't win the situation. While this is more of a joke to add pressure on the player in question, some teammates will actually start a vote in such cases.
Some CS:GO players, not only because of this, even go so far as to write themselves so-called "clutch scripts". By pressing a button, they can thus deactivate voice chat and music, send messages to the team or increase other volumes. With this, nothing stands in the way of magical clutch play, except for a small portion of luck.