The moon has lots of water, and oxygen, which is all you need. Almost rocks on over the ice that’s your fuel. Sure you could get all that on earth but then you have to haul all of that weight out of a gravity well, much easier to simply fuel a ship up with the limited amount of fuel it needs to get to the moon and then have its stock up somewhere with less oppressive gravity.

The question is, what kind of craft are you going to send and where to warrant Moon refueling as opposed to just another multi-stage rocket?

Literally anywhere in the solar system. Refueling on the moon is an enormous efficiency bonus, far apart from anything else not having a huge booster section massively decreased costs. In the long term you would probably build spacecraft on the moon and leave them in orbit of Earth and just have space shuttles handle the landing and taking off part. You could build enormous spacecraft with gravitational centrifugees since you no longer have to worry about landing mass or aerodynamics.