What is so bad about puns?

Many times I've heard of 'pun intended' or 'pun not intended', which I see as a form of excuse in the English-spoken world. However, I can not wrap my head around why are you constantly excusing/explaning something so innocent(?) as pun.

What I am giving off constantly saying 'pun intended'? What's the purpose of constantly saying 'pun (not) intended'?

We don't excuse puns in my native language, in my country, we just laugh it off.

It's not generally to excuse a pun, but to draw attention to it.

Sometimes "no pun intended" is an edit, or when the author/speaker realised an accidental pun (and in the case of writing decided to leave it in). In speech sometimes the accidental pun (or indeed any humour) may be inappropriate but it can't be edited out once said. In that case an excuse may be needed.

Sometimes puns are regarded as a low form of humour when planned, so "no pun intended" can mean "that just came out by itself" and may not be true. Other times they can be rather subtle and "(no) pun intended" highlights the wordplay and gives a pause to process it.

There are some cases in which it is used to excuse a pun: In technical writing the choice of terminology can be very restricted — everyday synonyms can mean completely different things. In this case the best phrasing may well lead to an inadvertent pun that is kept and the reader is asked to ignore it.

However, I can not wrap my head around why are you constantly excusing/explaning something so innocent(?) as pun. [...] We don't excuse puns in my native language, in my country, we just laugh it off.

I think you might be slightly overthinking how much we actually care about puns.

You've probably seen references to puns in internet comments, mostly because that's an informal context and because it's easier to make puns using written language. But in "the real world" you won't hear people referring to puns nearly as often as you do on the internet. This is a non-issue for us, just like it's a non-issue in your native language.

I will say that this depends entirely on context. Obviously you wouldn't include a bunch of puns in a speech you're giving at a funeral or at an important work meeting, but that's probably true in your native language as well.