Because the subject of the sentence is "one", not "the cats", and since it's a singular subject, the singular verb form needs to be used.
If you don't do so already, please read through the whole comment section before asking questions to see if your question has already been answered.
subject of thesentence is ONE,not CATS. what is your mother-tongue ? it must be the same. In french we say UN des chats DORT. not un des chats DORMENT. in spanish we say, Uno de los gatos DUERME, not DUERMEN. it is the same in german, portuguese ,italian etc.
I am afraid many of you are young people who did not study grammar in their own mother-tongues. WE had to. and it sure helps. At least for the Indo-European languages. I am still not good enough in ( or is it AT) ?Turkish,Swahili,Arabic,Japanese,Hebrew to say if the same rule applies in these langages.
Maybe it's because I'm from Yorkshire but never would I say "one of the cats sleeps." In real speech it would sound like one cat had been asleep several times and in "one of the cat's sleeps it maybe had a dream" One of the cats sleep. One of the cats is sleeping. One of the cats is asleep. Can't agree with "one of the cats sleeps."