Perhaps because English was still in flux, Elizabethans were fascinated with language, even the unschooled, and they delighted in puns. In Romeo and Juliet for example, as Mercutio dies he says, “Ask for me tomorrow, and you shall find me a grave man.” The pun, of course, is a play on grave, meaning “serious,” and grave, meaning “dead.” In King Henry IV, Part II, the Lord Chief Justice tells off the fat knight Sir John Falstaff, one-time drinking companion of Prince Harry, soon to be King Henry V:
Lord Chief Justice: Your means are very slender, and your waste is great. (“Waste” is a pun on the size of Falstaff's waist, that is, on his being fat. It is also a reference to the fact that he doesn't have much money because he's wasted it on drink, women, and gambling. Falstaff cleverly turns the word-play around.) Sir John Falstaff: I would it were otherwise; I would my means were greater and my waist slenderer.
So what did Elizabethan English sound like? In Shakespeare's day English was not the Received English of the upper and middle classes that we hear now in many plays and movies, nor the clipped cockney accent of London. Surprisingly, it was probably closer to the accents of northern England, Scotland, and Ireland, and some dialects of the rural eastern and southern parts of the United States. Language experts say that the inhabitants of Ocracoke Island, part of the Outer Banks of North Carolina, which was originally founded by Elizabethan sailors and their families, speak the closest to Elizabethan English to this day.
The reason to recapture the original pronunciation as much as possible is that the lines sometimes make much more sense when spoken in the original dialect. For instance, if you can imagine an Irish accent speaking the following lines from Julius Caesar you would not hear Cassius say, “Upon what meat doth this our Caesar feed, that he hath grown so great.” Instead, you might well hear, “Upon what mate doth this our Caesar fade … ” and now we have a clever play on words (mate, also meaning “friend” fade meaning “of lessening importance”).