A transfer request is a formal statement from a player asking to leave, usually made in writing to the club. It carries no power to force a move, because the contract still holds, but it can cost the player loyalty bonuses and it signals to buyers that the player wants out.
The contract is the point. A request changes nothing legally: the player is still registered, still owed their wages and still unable to play for anyone else. What it changes is the atmosphere, and clubs know that a public request damages the value of the player they are trying to keep.
It usually costs the player money. Contracts commonly make loyalty payments conditional on not requesting a transfer, so the request is a real financial decision rather than a free way to apply pressure, which is why many players prefer to let an agent apply the pressure quietly.
Clubs have their own version of the same manoeuvre: leaving a player out of the squad, out of the squad photo or out of the pre-season tour. Neither side can force the other to move, so both are trying to make staying uncomfortable enough that the other blinks.