I saw a production of Twelfth Night without ever having read it. Also didn't know the plot. And unfortunately I didn't ever get the gist of what was happening. I have a hard time imagining how other people are able to get the gist of the plays if they haven't read them beforehand.

It's not just the Shakespearean language. It's also the constant references to Greek and Roman mythology, via Ovid, that presents a real barrier. That mythology is no longer part of mainstream culture.