For example in The Times They are a-Changin' (Bob Dylan).
I suppose it's old language. Where does this prefix come from and what does it mean?
It is a very old use, no longer used in Standard English (whatever that is), but still to be found in many dialects.
The a- is originally a worn-down form of the preposition on. In OE a standard way of expressing a state was as a preposition phrase with on—on slæpe, for instance, "on sleep" = "in a state of sleep". We have quite a few common adjectives and adverbs beginning with a- which are derived from this construction: ablaze, afloat, afoot, ago, akin, alike, alive, aside. (Note that the adjectives are almost never used attributively, before the noun, but only in predicates: this is because they arose not as adjectives but as preposition phrases.)
The English progressive construction, which "recategorizes" an event as a state, arose from the same construction: it was originally on + the gerund of the verb, in ModE The times are on changing, "The times are in a state of changing*. Over time the OE participle and gerund collapsed into a single form, and on was apocopated to a; and eventually the sense of this construction being prepositional disappeared and the a was dropped, yielding the ModE progressive. But it lingers in remote dialects like the English of the Appalachians, and it's useful for songwriters like Dylan who seek an archaic, 'folksy' tone.
Incidentally, many linguists now believe that the progressive construction was borrowed by the Englisc from the Brythonic tongues they encountered. Modern Welsh still constructs the progressive with BE + the preposition yn + the infinitive of the verb.