As for usage, it is very easy:

Suspend the respective process with Ctrl-Z
Send the job to background using bg
Take away the ownership from the shell using disown
Start or enter your tmux/screen session
Run reptyr PID to attach the process to the current shell
It also has some additional useful features, such as TTY-stealing, which is documented in the man page.

Before compiling reptyr, make sure to check whether it is in you distributions repository. At the time of writing, this was at least the case for Fedora and Debian.