What does "so" in "I am so out of here" mean?

It is an intensifier; it emphasizes the emotional state of the speaker.

The implication of "so out of here" (and you do have to emphasize it when spoken) is that it's long past time, or that whatever is driving you away is obviously egregious.

Often we use swear words for this in English ("I am f---ing OUT of here"; "I am bloody well out of here" in UK English might be a minced version of this).

It's also an in-culture marker, used by a particular generation/culture exemplified by, say, the movie "Legally Blond".

Note: "[getting] out of here" always means "leaving this place", but the place might be metaphorical, as in "I am quitting this job".