Ss the name may mean; File Integrity Checking is checking for the integrity
of files against any reason that may have caused the change(intruder, buggy program, ........).

It Comes from the fact that when files (critical ones) change there
is a great chance that an intruder did it , and intruders always make sure that
the system operator don't know anything about the change
(some times add a ".file" in the /dev/ directory or change the "ps" and other
critical binary or conf files)

The only way to know about such thing is to periodically check for changes.