A day can be very busy and sometimes you have to deal with them all alone. Sitting at the desk in the nine o´clock in the night and thinking, "wish a nice warm meal appears right now" while you are poking knackered rucola and a few cherry tomatoes from three days ago as a dinner.
This is the moment you say, "Neko no te mo karitai (猫の手も借りたい)". Neko means cats, te means hands and karitai comes from the verb kariru (借りる) means to borrow. The last part tai expresses your wish, so karitai is want to borrow.
I want to borrow a hand of a cat because I´m too busy. That is the meaning of Neko no te mo karitai. You have too much work and you just need a help even from a cat although you perfectly know that it won´t help.
Usually it ends up like this. Your cat will just lie down on your side and do some stretch. You probably wish more that you were a cat rather than asking them for help.
So these are your dream but who knows, one day...
Or the most likely to happen is the cat comes and lies down over your book or your keyboard, the typical daily activity for the cat lovers. But you know, when you are really really busy, you feel like even the cat´s paw could be a help.
More about Japanese saying
More about Japanese cats