If I were to hire an external developer to customize my Salesforce org, how do they get access? Am I required to purchase an extra license just for them to do the initial setup, and make modifications every now and then?
If you are talking about Community/Portal users, then yes.
If you are talking about internal/platform users, then that's going to be tough. Actually, you're talking about a user who has to make configuration changes to your org, so that is, by definition, an internal user. You won't be able to buy a limited login internal license unless you have some serious spend with Salesforce.
I wish I had better news. You'll just have to deactivate another user while this contractor/consultant is working.