This phrase usually refers to men. Its background dates back to early 20th century American English and means a man who is seeking a female companion or trying to establish a romantic relationship with a woman.
It may be because cats like to eat fish, and men usually try to use some superb skills to do so. Attracting women, like cats when catching fish, requires some strategies. Therefore, "cats that eat fish" usually refer to men who have the talent to attract women, not women.