It appears that Animagi in general have closer affinity with animals. HP Lexicon picked up on this when discussing this same question:
Another factor presented later in the series are Voldemort's remarks to the Death Eaters on the subject of Wormtail in Goblet of Fire Ch 33, regarding his "curious affinity with rats". Implicit in those remarks is the information that Pettigrew could communicate with rats to obtain information involving rather complex concepts. It seems a reasonable deduction that Pettigrew's rat Animagus status is a factor in his ability to communicate with real rats.
Therefore it seems plausible that animal communication on a level above what normal wizards are able to do is common in Animagi.
There's no canon explanation but it may be related to this revelation about the nature of Animagi thought processes:
"They could tell that my feelings were less - less human, less complex when I was a dog...but they thought, of course, that I was losing my mind like everyone else in there..." (PoA, Ch 19, Sirius explaining how Dementors perceived him).
Add to that the fact of Crookshank being a half-Kneazle, who are known for at least SOME level of mind reading (they can tell good wizards from bad, and animagi from animals)