I have the zip file with everything in it, but there's just not a simple way to "run"? What all do I need? Do I need my own server or should they have their own?
No, there is no 'run'. Assuming your zip file includes a WordPress install and a database, then you'll need access to a web server and a MySQL installation. If you just want to run it on your own machine, you could install WAMP (if on Windows) or MAMP (for Macs) - these would give you all you need to get going. If you want to make this website live, however, you need a hosting company.