What does "a-" before a verb mean? [duplicate]

The a- prefix is a reduction of Old English an/on, meaning on, used to express progressive aspect.

English used to have more of a distinction between present simple and present progressive; what we now say as “the times are changing” was expressed in Old English as “the times change”. In order to emphasise the progressive aspect (the times are currently in the process of changing), you would have said “the times are on the change”. I believe this is related to some set phrases such as “on the run”.

The usage is still current in in Dutch as aan het (“Ik ben aan het lopen” = “I’m walking”). In English it’s now used almost exclusively in poetry—normally music, and folk music at that—since it provides a convenient way to fill out a syllable and tweak the stress pattern of the sentence. I don’t know of any English dialects that have it in everyday speech.

It is not true that English doesn't have any widely spoken dialects that use this.

Further, you leave out a big part of the story: "A-" before a verb was a prefix quite common in 16th C. English. It is still, today, quite common in Appalachian English, in the US, which is where Dylan no doubt took his influence.

It can mean "engaged in", as in "He's a-runnin! And fast!", or "She's a-birth, and there's no point in hoping she'll not."

It can also mean "motion to, into", as in "I'm going a-long", "I'm going a-bout", "I'm going a-round", "I'm going a-breast".

Being a colloquialism, its usage is largely regional, and so hasn't gotten enough attention to register on my personal "research radar" -- I, having lived in/come from Appalachia, find it rather intuitive. But my linguistic skills aren't sharp enough to describe precisely how the usage might work, unfortunately.

This is the sort of thing that, if you want to know how it's used, you've got to move to the place where it's spoken and hear it in speech for yourself.