The brevity and bluntness in his responses is pretty amusing. I'm surprised he so candidly rates his own abilities and the abilities of his peers - I feel like many people would dodge those questions in public interviews.
You don't know many Russians do you? ;)
In chess though you have a ranking, you have past games to compare. It would be silly for him to say "oh, I'm not that good" as there's a measured value which [possibly incorrectly] says he's the world number one. Similarly he spends a lot of his time assessing his opponents, their games, their characters.
I found it interesting that he said Kasparov can judge which opening a player will play taking in to account their mood. It would be nice to test that claim some how.
And besides, many players have their nemesis. Sheer numbers don't reflect that.
Eg. Anand used to be one of the strongest players in the world, but overshadowed, some say intimidated, by Kasparov before the latter retired.
Given Anand's rating, he performed quite poorly against Kasparov.
http://en.chessbase.com/post/computers-choose-who-was-the-st...
I remember there was a heated debate as for whether or not Crafty was a strong enough engine for the task.
I don't have all the links at hand, but it shouldn't be difficult to find various parts of this discussion.
http://en.chessbase.com/post/using-che-engines-to-estimate-h...
Authors of the research argued that contrary to what common sense implies, even if it Crafty is weaker than some evaluated players could have been, results are still legit.
Of course this approach doesn't give full justice to players - while some might have strived for winning by perfect play, others could deliberately play inferior moves just to create complications and provoke their opponents into making mistakes worse than their own, by getting them into time trouble, making them uneasy etc. (eg. Tal was famous for that).
A game - oversimplifying a bit - is won by the player who made less mistakes than the opponent, not by the one who made less mistakes on an absolute scale. So both strategies are valid as far as sport is concerned.
It's empirically checkable how close Carlsen is to Kasparov in skill. If Kasparov was significantly better than Carlsen in some ways, then Carlsen had to be his superior in others.
I'm reminded of a quote by one New England Patriots linebacker about another, when early in his career Jerod Mayo said (and this is approximate, although I'm sure the first three words are precise) "I'm pretty sure I can run faster than Tedy Bruschi, but he's still faster getting to a ball carrier than I am." That was a pretty big complement to Bruschi, about his quick instincts in judging a play.