Chris Beeston is an actor with Shakespeare's company who used to be an apprentice. He doesn't have much to do with the unmasking of Falconer—that was done by Mr. Armin and Widge. However, he does provide a clue when he and Widge talk about Leicester and stealing plays, which helps explain who Falconer really is.
Falconer is actually Simon Bass, the man who took Widge from Dr. Bright and tasked him to steal the play. Beeston says of Bass, "I never quite trusted him. One thing I will admit, he knew more about makeup than anyone else in the company." This foreshadows Falconer's eventual unmasking. Only someone with a great deal of skill at disguising others could disguise himself so well and for so long.
When Mr. Armin is dueling Falconer, he mortally wounds him. It's only at this moment that Falconer is willing to show who he truly is. He appears to pull away pieces of his nose and cheek to reveal that he is actually Simon Bass and not the scarred, terrifying Falconer. His skill with makeup—mentioned by Chris long before—is what allowed him to masquerade as the other man. Widge is very shocked, though Mr. Armin had already suspected that Falconer was actually Bass.