I Googled on the phrase. Is it from a novel called "The Naked Face?" Is this the context? "Where'd you get your medical certificate? From a veterinarian school? I wouldn't trust my fuckin' cats with you. You're a goddamn quack!" The speaker inserts the "f-bomb" into the sentence at a random place that doesn't make logical or grammatical sense, as an expression of anger. Logically, WITHOUT context, "f------ cats" would mean either that he is angry at the cats (likely) or that they are mating (unlikely). It is the verbal equivalent of pounding a fist on the table. You can put it anywhere in the sentence where the rhythm and cadence of the sentence make it effective. He's using "cats" with the idea that a veterinarian treating a cat doesn't require the same amount of trust as a doctor treating a human. "You're such a bad doctor that I not only wouldn't trust you to treat a person, I wouldn't even trust you to treat a cat."

A possible interpretation is that the person he is referring to is so untrustworthy, that he would not even let him look after his cats, even though cats are very easy to look after.

The speaker means to say that the person he is talking to is not very trust worthy and hence he would not leave his cats with the listener. The answer by Simon would be true if the speaker had used "I wouldn't EVEN trust my fucking cats with you" but since he did not use the word even so I don't think the answer is completely correct.

P.S. I said "you can put it anywhere in the sentence," but I meant that figuratively. Don't try to use this word when speaking English. It is a cultural minefield. It is true that it is now quite common to see it in books, and that you hear it all the time on the TV and in movies. Remember, though, TV and movies are often showing moments of strong emotion and high drama--MUCH more often than in real life. It is also true that it is a little old-fashioned of me not to spell it out. However, this is consistent with U.S. cultural norms for news broadcasts--on the news, if someone used that word they would "bleep" it, and when Pope Francis accidentally made a comical mistake in speaking Italian, newscasters described it by saying "he accidentally dropped the F-bomb." It is still a taboo word and it can get you fired or worse. (It can also get you blocked from some Internet forums).