When people say, "No pun intended," what does that mean?
That we're trying to be more serious and less playful that the use of a pun might demonstrate.
A pun is a form of word play that uses multiple definitions of words to create added meaning.
For instance, if I was an aeronautical engineer and my new experimental airplane was flying successfully over my head, I could say "Things are looking up" - referring to the plane being in flight and to the fact that my financial and business prospects are going to improve.
When we say "no pun intended", we're letting someone know that we have either accidentally stumbled into using language that could be seen as a pun but we didn't intend it to or that we can't think of any other way to say something and so are using what could be seen as a pun.
The reasons why we might want to "take out the fun of the pun" is that there are situations where we want to not be seen as funny or lighthearted - we want to be more serious. A pun can shatter that mood or meaning.