I think the statement is a little bit more nuanced than that:
"His over the board progress has been unusual, and throughout our game in the Sinquefield Cup I had the impression that he wasn't tense or even fully concentrating on the game in critical positions, while outplaying me as black in a way I think only a handful of players can do."
This isn't just "I got weird vibes" or something, this is the professional analysis of someone who has spent a lifetime analysing particular board states, the overall flow of the game, and the psychology of his opponents. He may have his hands tied in terms of what exactly he can say at this time, but the telegraph here is that he suspects cheating because of specific, observable factors in how the prior game(s) went down.
And those factors may ultimately be too subtle to be judged by anyone other than a jury of other top-tier professional chess players, but ultimately that doesn't matter, if it's enough to trigger a more thorough investigation then concrete evidence will emerge one way or the other and show Carlsen to be right or wrong on his hunch.