It looks like they have their own library for low level stuff (https://github.com/obsproject/obs-studio/tree/master/libobs), but for the most part rely on ffmpeg's library for all the platform-specific video, audio, etc. stuff (https://www.ffmpeg.org/about.html). If you browse the CMakeLists.txt files in the repo you can get a rough idea the kinds of libraries they depend on and use.