I am having difficulty understanding the purpose of increments in time control. What is the point? If it just adds more time to the game, why not simply make the game that much longer? I am aware that I am missing something fundamental here. Apologies for this complete newbie question!
It depends on what you mean by "increments". If you are referring to something like "playing a game in 10m + 4s", then the main reason is preventing that a player with a huge material advantage (v.g. king plus rook versus king) doesn't lose on time. With the increment of 4 seconds after each move is made, the player with the advantage will have no problem winning the position. Without increment, if the advantageous player has something like 1 or 2 seconds on their clock (and provided the other player has plenty of time, or just much more than a couple seconds) they'd probably lose on time -which usually is called "getting (dirty) flagged" in games without increment.
This is a somehow famous example of an online game in which Gata Kamsky was (dirty) flagged in a 3 minutes game without increment, with just one rook and king per side.
There is a big clue in the FIDE Laws of Chess in the section titled "Guidelines III. Games without increment including Quickplay Finishes" which deals with games or final parts of games with no increment in rapid and standard where one player is at risk of losing on time unjustified because they have a better or even winning position.
A modified version of this section was originally in the main body of the rules as what was the infamous article 10.2. Then it migrated to the appendices, specifically appendix G. Finally in 2017 in was moved to a new section of "Guidelines". The plan is that in the next iteration of the rules due this year (let's see if Covid delays that process) it will be removed altogether.
What this evolution tells us is that playing without increments, at least for standard and rapid, is deprecated. If you look at the guidelines you get more evidence of this.
III.4 If the player having the move has less than two minutes left on his clock, he may request that an increment extra five seconds be introduced for both players. This constitutes the offer of a draw. If the offer refused, and the arbiter agrees to the request, the clocks shall then be set with the extra time; the opponent shall be awarded two extra minutes and the game shall continue.
III.5 If Article III.4 does not apply and the player having the move has less than two minutes left on his clock, he may claim a draw before his flag falls. He shall summon the arbiter and may stop the chessclock (see Article 6.12.2). He may claim on the basis that his opponent cannot win by normal means, and/or that his opponent has been making no effort to win by normal means:
III.5.1 If the arbiter agrees that the opponent cannot win by normal means, or that the opponent has been making no effort to win the game by normal means, he shall declare the game drawn. Otherwise he shall postpone his decision or reject the claim.
III.5.2 If the arbiter postpones his decision, the opponent may be awarded two extra minutes and the game shall continue, if possible, in the presence of an arbiter. The arbiter shall declare the final result later in the game or as soon as possible after the flag of either player has fallen. He shall declare the game drawn if he agrees that the opponent of the player whose flag has fallen cannot win by normal means, or that he was not making sufficient attempts to win by normal means.
III.5.3 If the arbiter has rejected the claim, the opponent shall be awarded two extra minutes.
III.6 The following shall apply when the competition is not supervised by an arbiter:
III.6.1 A player may claim a draw when he has less than two minutes left on his clock and before his flag falls. This concludes the game. He may claim on the basis:
III.6.1.1 that his opponent cannot win by normal means, and/or
III.6.1.2 that his opponent has been making no effort to win by normal means.
In III.6.1.1 the player must write down the final position and his opponent must verify it.
In III.6.1.2 the player must write down the final position and submit an up-to-date scoresheet. The opponent shall verify both the scoresheet and the final position.
III.6.2 The claim shall be referred to the designated arbiter.
The first thing to note is that the very first way of handling this situation is to introduce increments!!
If you read on you see why this is so although it is not explicitly mentioned. That is that the result of the game is decided by the arbiter making a decision and not by the play of the players. This is regarded as being both unjust and unfair to the players and unfair to the arbiter who has to take responsibility for deciding the outcome of the game.
The principle function of the arbiter is spelt out in article 12:
Article 12: The role of the Arbiter (see Preface)
12.1 The arbiter shall see that the Laws of Chess are observed.
12.2 The arbiter shall:
12.2.1 ensure fair play, ...
It is not the job of the arbiter to decide who wins, loses or draws, yet that is what the Quickplay Finish rules call upon the arbiter to do.
You may be wondering why I have included section 6 and bolded it. Section 6 applies when there is no arbiter present at the tournament. This is not ideal and is not permitted for high level FIDE tournaments but is allowed in things like local leagues which are FIDE rated. What happens here in such a case is that the game stops when the player makes the claim, the position is written down and the scoresheet with the moves and position is sent to the off-site arbiter. They then make a decision. If they reject the claim then the player making the claim loses the game. If there is doubt in the mind of the arbiter then the benefit of the doubt is given to the other player, the player who is supposedly trying to win.
That is why in the next iteration of the rules this is section is likely to be removed and games played without increments will be sudden death as it currently is in blitz.
It ensures that you'll always have some time left after you move so that you won't lose a game "on time", i.e. as long as you move within the incremental interval. For example, if it were a 10 minute game with 5 second increments, and you got down to a point where you only had several seconds left on your clock, as long as you could continue to move within the 5 seconds for each move, you would never run out of time. Of course, by rushing you might not be making the best moves and could lose that way if your opponent capitalized on any errors you might make. To answer your direct question, it adds time to each move, not just to the entire game, which is a totally different thing. If you just added time to the game, you could still run out when you got to the time limit since you wouldn't be able to move fast enough to avoid that.
I have 2 ways to respond to this post
3min + increment 2sec is not exactly equal to 4min20sec + increment 0sec, but 4min20sec is the benchmark to estimate how long a game will take to classify 3min + increment 2sec as blitz/bullet/rapid/hyperbullet/ultrabullet/classical/slow rapid/fast rapid based on that an average game is supposedly 40 moves.
However, a game may not be at most 40 moves. It could reach 60 moves. In particular...
...a no increment game of 60 moves may end largely in part of whoever/whomever can move faster and find/avoid checks faster.
My 2nd way to respond to this post is that I think usually people will ask instead why people still don't play with increment after Fischer introduced increments (to chess and chess960...and I guess to go, shogi, xiangqi, etc. Wonder how those games were affected by Fischer's increments). So, I think the question of why people play with increment can be answered with Q&A's about why people play without. Eg this reddit post:
What's the appeal to playing no increment chess? https://www.reddit.com/r/chess/comments/hnj8ot/whats_the_appeal_to_playing_no_increment_chess/
It was done to make Fischer happy. He thought that if he was in time trouble that he should still have xx seconds to move.
Others of us say that chess is a timed game and if you get into time trouble then you deserve to blunder or see your flag fall.
Increments make the game much longer and for speed chess totally distorts the time control. My friend likes to play 15min game with ten seconds added per move. I say that is like postal chess as I prefer to play at ten minutes dropdead game over after you used that much time.