The expression is slang, but not really humourous although it can be used in such a context.
Actually, it is often used as a negative. Say a friend just got dumped and he needs a little support and after 3 months he's still living in your spare room:
« Il s'est incrusté dur là! »
The idea is that someone entered somewhere and stayed when not invited to do so for so long. In this case, “dur” (hard) suggests that he stuck around.
The first meaning of incruster is inlay (as in art, as in digging a small hole to place a jewel on a crown).
"sneak into" is certainly a good translation for an animal, especially since in such cases it is not likely to bother you on purpose (unintentional) and also those animals are probably not going to stay for long.