I'm Italian so I don't know English very well.
While listening to Bob Dylan songs I've heard some strange use of progressive tense (is that the correct term?), the title of this question is one example. Why "the times are a-changing" and not "the times are changing"? I heard other examples (always in Bob's songs), but now I cannot find any.
Is this some sort of ancient English? Slang? Metric trick?
"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".
It is not today used in formal English, but it is an archaic usage from the 16th C. English (as one sees in the words "around", "about", and "abreast"), so it is not an "Americanism". It is an archaic form of English that survives in America (and with consideration for the powerful influence this prefix worked over our prepositions, I'd suppose it survives in parts of Britain, as well -- Scotland, perhaps?).
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.
It is archaic or dialectical. You would hardly ever use it in modern, spoken English, so I wouldn't worry about it.
When it is used, it's almost always in highly idiomatic ways, I can't think of many... but here's one that sprang to mind: "Hurry up people, the clock's a-ticking!"
I feel like Bob Dylan probably used it for metrical effect... certainly I don't think you'll find anyone who could find any difference in meaning between Bob's lyric and just "the times are changing".
Indeed, this sort of usage is highly idiomatic. I have the sense it is more typically seen in Southern, Southwester, rural, and folksy dialects. (I'd say you'd see this construction correlated with swallowed final "g"s--runnin', jumpin', etc.) Compared to "changing", then, "a-changing" evokes a rustic, or perhaps populist sense, which is likely the tone that Dylan is trying to achieve by using this form.
Off the top of my head, I'd say that this usage is limited to progressive tenses (with the present active participle). However, I wouldn't completely rule out seeing this occasionally with a past participle.
One rule of thumb when I studied Latin verse was to never assume that the reason for a particular construction was to "make the meter work out". I'd say the same guideline applies here. There are enough ways to say things using varied meter that it is customary to give the poet (or songwriter) credit for having some deeper reason for using the words that (s)he chooses.
