I disagree with most (though not all) of the points mentioned in existing answers, and find them to arise from a rather naive and unsophisticated understanding of the game. I will first begin by explaining why some the points already mentioned are NOT the answer to the question.
One of the existing answers presents the answer as saying that prior to the modern era, people believed that chess was based on rules like controlling the center, whilst the new advances involve the "initiative" and "exchange sacrifice" and an understanding of "rule independence". This is totally ridiculous, because players all the way back to Paul Morphy, one of the earliest 'great' players of the game, played extraordinarily "romantic" chess which was all about sacrificing pawns for activity. Tal is also another famous, more recent example. To the contrary, modern players are more likely to play slow, maneuvering chess compared to the "romantic" era of Morphy and Andersson, like the countless Berlin defences (popularised by Kramnik in the 2000 World Championship against Kasparov) and Ruy Lopez and Giuoco Piano slow maneuvering chess seen nowadays at top level.
The books by Nimzowitch and Watson, and indeed any chess book in general, merely represent the points that the author thinks to be instructive for the reader and valuable to include in the book. The idea that this can be taken to be a summary of existing knowledge in chess, many aspects of which (e.g. a strong chess player's "intuition") is not even possible to really write down in a book the first place, is rather naive. All of the points mentioned in Nimzowitch's book was surely already understood to some depth by Paul Morphy, and even by any average 2000 rated player in 1900 or in the modern day. For example, the "exchange sacrifice" is an idea that even 1000 rated beginners have probably heard about, and these surface-level words and phrases have no relevance to the process of thinking at top-level chess. This has absolutely nothing to do with what Garry Kasparov was talking about when he mentions the advance in chess knowledge.
The real difference primarily lies in opening theory.
Prior to the age of engines, strong players who are looking to gain an advantage as white in the opening would need to discover these ideas for themselves through extensive analysis and great creativity. In complicated positions, it may often not even be clear whether or not a certain idea is good; evaluating these positions is a very difficult and sometimes tedious task (in terms of analysing lines). Nowadays, if Magnus Carlsen were trying to prepare for a World Championship match, he (or his team of seconds) could simply look to the engine for reliable analysis of lines and there is more or less no chance of making a severe error in analysis that could cost him the game by walking accidentally into a lost position out of the opening. Of course, there is still a lot of work to be done, in the sense that the opponent has access to engines too so it is difficult to come up with creative ideas to surprise the opponent, but it is still a great advantage in terms of quality of analysis compared to the pre-engine era.
Additionally, from the decades of games that have been played since Kasparov's time, chess theory in various popular openings have developed and improved, so that the top players know a lot more details about various lines than Kasparov would have been able to know. This is simply because more ideas would have been played by strong players, winning some nice games, and everybody (in the top-level chess world) takes notice and learns the lines too. Some topical examples includes detailed analysis in popular lines such as the Najdorf and Berlin, as previously mentioned. (Though I will admit that I am not strong enough to give detailed analysis of all of the intricacies of these lines and how they evolved over the past 20 years, and you should probably not believe anyone who claims to do so unless they are a strong GM.)
Another concrete example is that of openings like the King's Indian Attack, King's Indian Defence, and Queen's Indian Defence, all of which have been borderline refuted by engines. For example, in computer championship games, the side which plays the opening has an overwhelming disadvantage and loses a large proportion of the games. This is true even for openings like the Caro-Kann and the French defence. This has led to a drop in popularity at top level for many of these openings, when they had been popular even just a decade ago. For example, the famous game Gelfand-Nakamura from 2010 ("Nakamura's immortal") is a great example of a tremendously complicated middlegame where Nakamura won in the King's Indian Defence with the black pieces, where a critical position involving queen sacrifices on consecutive moves occurred.
[Title "Gelfand vs Nakamura, World Team Championships 2010"]
1. d4 Nf6 2. c4 g6 3. Nc3 Bg7 4. e4 d6 5. Nf3 O-O 6. Be2 e5 7. O-O Nc6 8. d5 Ne7 9. Nd2 Ne8 10. b4 f5 11. c5 Nf6 12. f3 f4 13. Nc4 g5 14. a4 Ng6 15. Ba3 Rf7 16. b5 dxc5 17. Bxc5 h5 18. a5 g4 19. b6 g3 20. Kh1 Bf8 21. d6 axb6 22. Bg1 Nh4 23. Re1 Nxg2 24. dxc7 Nxe1 25. Qxe1 g2+ 26. Kxg2 Rg7+ 27. Kh1 Bh3 28. Bf1 Qd3 29. Nxe5 Bxf1 30. Qxf1 Qxc3 31. Rc1 Qxe5 32. c8=Q Rxc8 33. Rxc8 Qe6 0-1
Despite this incredible win, which you might expect means that the popularity of the King's Indian Defence would surge amongst top level players, if you check the database, the last time Nakamura has played this opening as Black in a classical game was against Lenderman in the US Championships 2018 which ended in a draw. He has rarely played this even in rapid and blitz in serious tournaments since 2019. You will see similar trends if you look at the openings played in tournaments like the Candidates with a surge in popularity of reputable openings like the Ruy Lopez and fall in popularity of these imbalanced (and arguably engine-refuted) lines.
Hypermodern openings such as 1.b3 (Larsen's opening), 1.g3 or the Grunfeld offer an example that is less recent but maybe easier to understand. These openings were deemed to be dubious since they violated classical opening principles, opting to control the centre with long-ranged pieces (like a fianchettoed bishop) rather than pawns. The Grunfeld, for example, relies on active counterplay and undermining White's strong (but overextended) pawn center, where Black actively gives up center control. But once people such as Bent Larsen (a former 4-time World Championship Candidate) started playing moves like 1.b3, top players realised that hypermodern ideas were not as bad as once thought; indeed, these have now become popular openings in online blitz (with Nakamura, the currently FIDE blitz rated #1 in the world frequently playing 1.b3 in online blitz) and even in serious classical OTB tournaments (Maxime Vachier-Lagrave famously has the Grunfeld in his repertoire). To be clear, these openings have been known since more or less the 1920s, so Kasparov definitely wasn't talking about them when referring to new knowledge of modern players. Nevertheless, they offer an illustrative example of the nature of advance of chess knowledge that is easier to understand for non-GMs.
TL;DR What do modern chess players understand about the game that previous generations did not? The answer is detailed intricacies in many different lines of opening theory. This is why it is reasonable to think that Magnus Carlsen, or maybe even Fabiano Caruana, or Levon Aronian, or Ding Liren, or Ian Nepomniachtchi, at their peaks would likely (or at least have a real chance to) best Garry Kasparov at his peak. It is simply an unfair comparison because Kasparov did not have the advantage of knowing these few decades of opening theory that have developed, aided by modern engines, since his retirement. These concrete lines, variations and details are the difference, not some vague claims of "inventing a new style of chess" or "rule independence", which are total nonsense.