I was wondering about this question for a while. Why did Bobby Fischer resign his title?
Bobby Fischer did not resign the title, although it may have looked that way.
As the New York Times reported at the time:
The International Chess Federation stripped Mr. Fischer of his title and gave it to the challenger, Anatoly Karpov of the Soviet Union, because the 32‐year‐old American failed to meet the deadline for formal acceptance of federation rules for a championship match this year.
To add further to the confusion according to Wikipedia:
In response to FIDE's ruling, Fischer sent a cable to Euwe on June 27, 1974:
As I made clear in my telegram to the FIDE delegates, the match conditions I proposed were non-negotiable. Mr. Cramer informs me that the rules of the winner being the first player to win ten games, draws not counting, unlimited number of games and if nine wins to nine match is drawn with champion regaining title and prize fund split equally were rejected by the FIDE delegates. By so doing FIDE has decided against my participation in the 1975 World Chess Championship. Therefore, I resign my FIDE World Chess Championship title.
The delegates responded by reaffirming their prior decisions, but did not accept Fischer's resignation and requested that he reconsider. Many observers considered Fischer's requested 9–9 clause unfair because it would require the challenger to win by at least two games (10–8). Botvinnik called the 9–9 clause "unsporting".
So, FIDE said "We don't accept your resignation. You're fired!"
Basically, just as in 1972, Fischer made demands that the rules be changed. FIDE and Karpov accepted one of his two proposed rule changes but rejected the other.
According to the NYT:
The federation last month agreed to his demand for an unlimited match, instead of one with a 36‐game limit, with draws not counting and the winner being the first to take 10 victories. However, it rejected his demand for having the champion retain the title in the event of a 9‐to‐9 tie.
Fischer was essentially not just demanding draw odds but, because of the other stipulations, was demanding that his opponent had to beat him by 2 clear games to take the title. This was rightly rejected as giving him an unfair advantage. FIDE gave Fischer an original deadline to accept the conditions and then when he didn't reply they gave him an additional day. When he still didn't reply they did the only thing they could and defaulted him.
It sounds as if Fischer was just being stupid and some people said he deliberately made an unreasonable demand because he was frightened of Karpov but that fails to take account of Fischer's poker-like approach.
In 1972 he made demands that put the match at risk and they paid off.
He might reasonably have expected the same to happen in 1975. The key point being that there was $5 million dollars at stake if the match went ahead, with the winner getting the larger share but the loser still getting a large sum of money. Granted the Soviet chess federation would have taken a large chunk of Karpov's share but Karpov would still have been rich. No match, no money!
Bottom line Fischer's gamble didn't pay off and his bluff was called.
Fischer wanted to play under a first-to-ten-wins format. This was the format of the first world championship match between Steinitz and Zukertort. The argument for this format is that it forces whoever is leading in the match to continue to play for a win. For comparison, in 1972, the Fischer vs. Spassky match was first-to-12.5-points. Therefore once a player is in the lead, they can just trade some pieces, draw some games, and coast to the title. This is (arguably) what happened in games 14 to 20 of the 1972 match.
The main argument against this is that it is impractical, and matches can be of unlimited duration. This can be dangerous to the health of the players, as happened in the World Chess Championship 1984. Furthermore, first-to-ten-wins gets more and more impractical as players get better (which they do, since chess theory improves over time). In the Steinitz vs. Zukertort match the championship was decided after only 20 games, but Fischer vs. Spassky 1992 took 30 games. This doesn't sound bad but neither Fischer nor Spassky were top players in 1992, and earlier in the World Chess Championship 1984, in which one "only" needed to win six games to win, there were still only 8 decisive games in 48 games played. And if you are masochistic enough to pit two top chess engines against each other from the starting position on strong hardware & at long time controls, you could very well end up with 100% draws.
In the lead-up to the 1975 match, Fischer had sent his demands for a first-to-ten-win format to FIDE. There were heavy deliberations within FIDE, with the federation ultimately voting against an unlimited match. Fischer refused to defend his title as a result. You can read more about the episode on Fischer's Wikipedia page.