For dance's significance, I like Werner's comment best.
For the verb form, Vader had it, almost. It's the contracted form of "I would have" which can be "I would've" in writing or speech, and also "I'd've" primarily in speech, and in this case. Thus, the correct rendering (grammatically speaking) is "I'd've had to miss the dance."
Note that this is mostly spoken English and rarely written, except as quoted or reported speech.
It is pronounced *exactly* as if it were "I'd of had to miss the dance", but this latter form is wrong. Note however that, pop and country music affect a style of speech which is intentionally common, and thus stay away from too-exact grammatical niceties, like this one. So, if Garth wrote the lyrics, he may very well have written, "I'd of had to..." either knowing it was wrong all along, or maybe even not knowing it.
fl: your comment
> Educated native speakers do not 'gerne' say 'I'd of had to'
is right, sort of. The problem is, in order to determine the truth of this statement, you must be able to see what's inside the head of the speaker. Had
you written 'do not WRITE 'I'd of had to' then I would agree with you without qualification. But since the pronunciation of the incorrect and correct versions are identical, there's no way to know, really. However, given that they're educated, yes, I agree with you, they don't say it (read: intend it) that way.
> Tatsächlich ist es 'av', eine Slangform von 'have'.
Have no idea what you mean. There is certainly no such form in the USA, slang or not; possibly elsewhere there is.