The phrase 'No pun intended' is a truncated version of the sentence 'There is no pun intended'. It's like saying 'No offence!' rather than 'I didn't mean any offence', or 'My pleasure!' rather than 'It was my pleasure'. We can often truncate well-known phrases in this way.

I agree with La Liseuse that "No pun intended" is short for "There is no pun intended." You would not say "There are no puns intended" as you would typically be referring to only one pun. Thus, "No puns intended" would not be correct. Also "pun's" with an apostrophe reflects possession rather than the plural "puns", so writing "pun's" would also be incorrect for this reason.

When someone says no pun intended, however, it is when they say something that could be taken to have two meanings; an obvious one, and a slightly "off" interpretation; and they want you to know they are not making an intentional play on words and want you to take what they said to be the obvious meaning. so when they say 'pun' it refers to the joke within the sentence.