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go

9. Informal To say or utter. Used chiefly in verbal narration: First I go, "Thank you," then he goes, "What for?"

American Heritage® Dictionary of the English Language

Examples that show the usage extends to describing sounds that people, animals or things make:

I fly into JFK

My heart goes boom boom boom

I know that customs man

He’s going to take me

To that little room

from Paranoia Blues by Paul Simon

And the colored girls go

Doo doo doo doo doo doo doo doo

from Walk on the Wild Side by Lou Reed

Verbs with very broad meanings like do or go get the sense of "say" in many languages. English uses "go" to mean "say" in very informal speech. There are children's songs about animal sounds that use it in this way.

"Say" applies to speech, and dogs, cats and birds can't speak. Which means "say" is the wrong word.

What would you place there instead? "Vocalizes"? Works for dogs, cats and birds, but what about a crickets and bees? They don't vocalize. How about "Emits" as the catch-all for animal noises? Way too collegiate.

There are often more specific verbs: The dog barks "Woof". OK so far. The bird tweets "Tweet". That's awkward. The cat coos "meow". Well, now we're adding meaning, since cooing suggests a different mental state than mewling or chirping (just read any mystery which involves cats trying to help).

This is a linguistic Gordian knot.

A relatively common, generic word like "Go" is about as good as this situation will get... and it's not so bad. It adds a bit of stylistic color to the language, and certainly specifies a level of formality (or to be more precise, informality). I wonder how a proper British butler would speak of it.

Dog goes "woof."
Cat goes "meow."
Bird goes "tweet."

Is there some specific meaning for "go"?

No, it's just a broad term marking an action. Much like "A dog makes woof."

And "A bomb goes boom" is most definitely not a task of speaking - in fact, the use of go(es) is so unspecific, that here it could mean the sound as well as the explosion or the destruction.

Say, in contrast, includes the very specific meaning of speaking (or text to be read). It's usually not including animal sounds - unless in some transferred, poetic context (let the wind speak for example.)

The duality is found in next to all languages with Germanic roots - already found in 12th-century texts.

It could have as well been used in the upfront question "What does the fox say?" as "How does the Fox go" except that usage of 'say' does explicity ask for a sound/word 'spoken' - not to mention that it's everyday kids' language :-)

I don't think that Ylvis paid any special attention to what verb they used for their narration of animal sounds (unless they aren't good English speakers - can anyone confirm/deny?). It's simply a different way to say "says", albeit less formal. I doubt there was any special intent to create comedic effect out of "dog goes woof" beyond the rather ridiculous idea of including nursery-rhyme-esque statements in their lyrics. The verb "to go" here is nothing special. "Dog says woof" would have been just as acceptable.

The MacMillan Dictionary gives many definitions of "go"; one of them is "to make a particular sound, especially the typical sound of a particular animal." "Go" would also apply to the sound of an inanimate object: for example, "toot goes the whistle" or "clang goes the bell." In those cases "says" might be understood, but "goes" is more natural.