The premise is simple. You leave food and toys in your yard to attract cats. You buy food and toys with fish. Cats come to visit and leave fish as gifts for you. You can then purchase more food and toys. And just like in real life, you can take photos of the cats when they come and keep them in your kitty albums. Neko Atsume is utterly pointless, and completely charming.
Here you can see that I really care for my cats, as I've put out expensive food with bonito for them.