Erika -
I also sell on Abe's Market (www.abesmarket.com) where I do drop shipping. Basically when they get an order for my products, they send me an invoice to my online sellers account where I can view the order. I get the order together and weigh it. Abe's Market pays for priority shipping so all I have to do is weigh the final box, and type it into this shipping calculator that they have. Their site generates the label which I then print out onto label sheets they send me, and I pop the package into the mail the next day!
If you imagine that all of the shops on Etsy were connected where the customer checked out all at once but then the different sellers got sent individual invoices and sent all the products separately. That is basically what drop shipping is. Hope that helps explain it a little better!