I was wondering what processes exactly use the
cache directory inside my theme's directory. More specifically, I am wondering if I should add any of those files to my git repo.
If you use someway "compilable" WordPress theme (are they exist?!) cache-dir may contain theme-files, processed by some tools and prepared for using by httpd-server of site.
As all and any artifacts, constructed from versioned data, these files can and have be excluded from versioning - everybody will be able to rebuild these files from versioned source files