Primarily what is the Reset Skip, what is it good for, and how is it performed?

Reset Skip is a general name used for any setup on an island that allows you to reset, and either retain some items or skip a certain portion of the island. This saves time because you don't need to get those items or do that part of the island. The performance is different for the different islands. There should be instructions for some of them in the guides. Otherwise, pm Jason, who I believe runs and discovered most of them.

a good example is the one for 24 carrot island, where you get the robot hat and save it as a costume, so you can wear it before you get it, letting you skip a good portion of the island

I've tried to do that, but when I reset, I lose the hat. How do I do it?

If you have the hat saved in a costume, open up a customizer tab, either by customizing off an npc or by opening a store outfit, then open your closet (the hanger button in the bottom right), and select the costume with the hat, then equip it.