It's hard to tell what "break" means here. Because this is an actress, my best guess is they're talking about break in the sense of a career opportunity. From Oxford Dictionaries:
break (n) - an opportunity or chance, especially one leading to professional success
The term "her big break" is sometimes heard when referring to actresses and other performing artists. It means that the person used to be unknown in the industry and was enjoying little success, and then something happened, the "big break", and after that they became famous and successful.
I'm guessing that Meryl Streep used to act in a play at this Eugene O'Neill Theater back when she wasn't famous, and now that she is famous they are presenting her with an award.