Most MUD applications enable you to keep a log of everything that happens in your sessions for later review. The logs are usually stored in text format so you can later import them into other applications, and they sometimes make very interesting reading after a session is over.
To log your session, with most application you first set your log folder, then turn the logging setting on. After you log out, you can import the text log into a word processor to read, take extracts, or format for some other use.
Note that it is considered very bad netiquette to post on the Internet or otherwise make public the logs of private MUD sessions.