A good question. The fire escape is something that could be a literal escape, in the case of a fire. However, it is also a symbolic escape in at least three other important ways.
First and most simply, it allows one character to escape from the others for a while—to not be around them, and to be…separate. That family is uncomfortably wrapped together, and that's necessary.
Second and a bit more symbolically, while the family can go out there, they go out into an alley. It isn't much of an escape compared to their grand dreams, just as the jobs they hold or try to hold don't much compare to the mother's faded glory.
Third, and most symbolically, it serves as kind of a stage for dreaming. Look at Tom's opening lines and you'll see this.