My main question: Do you, as the app developer, even need to construct these HTTP requests or do you use their SDK or something completely different and I'm just not getting it?
This is entirely up to you, the developer.
If you want to roll your own and construct your own HTTP requests, you certainly can. You almost certainly will still want to use a pre-packaged OAuth library, as OAuth is not trivial to implement.
However, you could also certainly use an existing code library/DevKit too, in which case the library/DevKit will construct the HTTP requests for you.
The DevKits should contain example code to show you how to actually do this stuff, so that might be your best place to start.