What is /proc? Can I safely remove the large files there?

The /proc filesystem is a memory image of each process; it's a
virtual filesystem that occupies no disk space.

/proc is used for programs such as ps(1) and top(public domain)
and all the tools in /usr/proc/bin that can be used to examine
process state.

The only way to remove /proc is to unmount it, but that will break
a lot of useful programs.