It seems to be an artistic license, an incorrect form of Times are changing. Incorrect because of the two subjects (not using and/or), times and they, for a single verb.
This is an example of "left dislocation", a common (though often informal) construction in English. The "normal" sentence would be:
The times are a-changin'.
Huddleston & Pullum write (with some light editing by me):
A dislocated clause has a constituent, usually a noun phrase, located to the left or right of the nucleus of the clause, with a linked pronoun or comparable form within the nucleus itself. Compare:
i Her parents seem pretty uncaring. [non-dislocated version]
ii Her parents, they seem pretty uncaring. [left dislocation]
(Huddleston & Pullum, The Cambridge Grammar of the English Language, pg. 1411)
Note that it is common to include a comma between the dislocated noun phrase and the nucleus. I'm not sure why Dylan didn't do so.
Is it how people say something is changing? Does someone using this sentence make an explicit reference to the song?
People do sometimes use this exact phrase, and in such cases most listeners (or readers) understand that it is a reference to Dylan's song, which is very famous.
Is it limited to spoken language or used in writing?
According to Language Log,
Left dislocation is certainly grammatical in English. Up to 1500 or so, roughly one in every 100 or 200 sentences had this form, even in formal writing, and a similar frequency of use continues in spoken English to this day. Over the past few centuries, the frequency of this construction in standard written English has been declining, and it's now quite rare except in archaic styles, in representations of speech, or in informal styles that use spoken-language norms.