I am a chess professional, although more as a teacher than a player, so I have a lot of experience in this field. As caro_douglos pointed out, it slows down the initial reaction to opponent's moves, which is one reason why it instantly improves performance for new players. It forces the player to settle into a certain rhythm of play which prevents silly mistakes.
It is also permitted (and encouraged) to write down the time on the clock after each move, and to record all draw offers, both of which are valuable information during and after the game. If one of my goals is to grind down my opponent's clock in a complex position where I have the initiative, seeing how much more time he is using on average gives me an idea of how long I need to maintain the tension to gain a sufficient time advantage. For example, in a game where both players start with two hours, entering the endgame with 1 hour versus my opponent's 15 minutes is an enormous advantage, even in a position which should otherwise be drawn with perfect play.
Also, in some time controls, players gain extra time after a certain move (generally the 40th), so it can be very important to know exactly when you're going to get that extra hour. These time controls are less common than they used to be.
I have students as young as 5 years old learn how to keep notation after only a few weeks of learning chess. It is a very important skill.
When it comes to studying for an upcoming tournament match you will know who your opponent is so you have the ability to study up on their previous play. Maybe they play a certain style against your mainline go-to that you can exploit.
It is also permitted (and encouraged) to write down the time on the clock after each move, and to record all draw offers, both of which are valuable information during and after the game. If one of my goals is to grind down my opponent's clock in a complex position where I have the initiative, seeing how much more time he is using on average gives me an idea of how long I need to maintain the tension to gain a sufficient time advantage. For example, in a game where both players start with two hours, entering the endgame with 1 hour versus my opponent's 15 minutes is an enormous advantage, even in a position which should otherwise be drawn with perfect play.
Also, in some time controls, players gain extra time after a certain move (generally the 40th), so it can be very important to know exactly when you're going to get that extra hour. These time controls are less common than they used to be.
I have students as young as 5 years old learn how to keep notation after only a few weeks of learning chess. It is a very important skill.