The word "escape" is used repeatedly in the play, and the fire excape is the physical manifestation of the escape which is central to all the characters. Tom escapes to the movies, before literally escaping the clutches of his family by joining the Merchant Marines. Amanda escapes to the "Blue Mountain" of her youth. And of course, Laura escapes in her glass menagerie.
Significantly, there is only one entrance and exit to the apartment that we ever see. For Tom, the room of his memory consists only of the living room, a half-formulated kitchen and bedroom, maybe, and the fire excape. The fire escape is his route to the outside world, and Tom begins and ends his story there.
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.
For Tom it was an escape to adventure. Tom wanted adventure which is why he left to join the Merchant Sailors. It was an escape away from home.