"Keep in touch" is a common phrase to use when you're permanently saying goodbye to someone, like when you:
- leave a job
- graduate from a school
- move to another city
To "keep" doing something means to continue to do it. Being "in touch" means that you communicate with someone. So "Keep in touch!" means "Continue to communicate with me!"
However, people say "Keep in touch!" so often that sometimes it doesn't really mean much. If someone tells you to "Keep in touch", they probably don't expect you to call or email them.
If you really do want to "keep in touch" with someone, you might need to say or write it in a different way like this: