Rar is pretty proprietary–and not supported under Engrampa. Apparently one has to pay a licensing fee–but the developer makes uncompress-only clients available for free.
Have you tried RARLAB’s official package for Linux? It is available here:
What is the program called?
When I installed it it said it was available in the repositories and suggested it be installed there but did not say the name of the program. How do I launch it? When I right click on a rar file I don't see any new choices for programs to use with the file.
UNRAR There are two versions in the repositories one free and one non-free. Both are older than the version on the website.
Manually installing files from debs is, for me at least, a last resort because it can break update dependencies. However, you are unlikely to ever have another program depend on this one.
Have you eliminated @pavlov_kairis comment as a cause? That is a good catch to notice the directory in the screenshot. None of the archive programs would work if they do not have write access to the directory.