A few different ways. If you ask to do it at the time of purchase, you will be given a form to fill out (believe it is the same one as if you were shipping to resort) and then be charged a set rate based off the total transaction cost. They bag it up, staple it closed, and then it shows up at your house later. Pros, you don't have to worry about anything, if something shows up broken or missing it's their fault and they are usually great about fixing/replacing it. Cons, it can be really expensive if you are shipping smaller, more expensive things home. Tip - You can save up your purchases (and receipts) and have them do it all at once instead of doing it multiple times. Be prepared to wait as the store basically needs to return everything, and then resell it to you in a single transaction.
You can ask your front desk to ship it for you. They will ship it via UPS, FedEx, USPS, whatever you want that they have access to. Pros, you only pay for whatever the shipper charges, plus a handling fee. So shipping a small expensive item will usually be cheaper this way. You can also ship things that Disney won't, such as food or stuff purchased from 3rd party stores. Cons, you need to find a box and packing material yourself. Your resort store may have something, they may not.
You can shop via the Disney Parks App/Merchandise Hotline. Pros, you can shop from all the Disney O/O stores on property at once and have it shipped home. And if you shop via the App they do occasionally run sales or free shipping (and you can apply AP or DVC discount if your email is the same for the various accounts). Cons, same problems with shipping, you ship per transaction cost, not size or weight.