This is, in a nutshell, how a Fringe Festival works. First thing is, no Fringes are identical. Some utlize a lottery system, some it's the first 100 (or 500 or whatever) plays a producer submits. Some festivals you submit the play and a selection committee decides.
Once your play has been chosen, it is up to the producer [from here on out will be referred to as "you"] (in some cases the playwright) to find or serve as the director, assemble the cast, and orangize the rehearsals. You usually know months in advance if your show has been selected. Once you are in the hosting city, you are usually given one or two rehearsal periods to run through the technical stuff (your play has projections on the back wal and sound effects that go through the sound system--this is when you run through that with the equipment on hand.) This is also when you check to see how long it will take to set up and tear down your set. Fringe Festivals usually give performers 45 mins to an hour, with a half hour to set up and a half hour to tear down (sometimes less). This is strictly followed.
Most Fringe Festivals serve only as host. You, as a producer, pay the festival a set amount (usually depending on a) the size of the venue you want (hint smaller is cheaper) and b) length (some will offer a 90 min block but you will pay more) and c) number of shows you want to perform.) Attendees then usually pay for a Festival pass (maybe for the two weeks or whatever of the festival or for the day or weekend) and then wander or peruse the program for a show that catches their eye. Producers are hawking their shows, handing out flyers, talking to reporters, reviewers, etc. Doing everything in their power to get people to attend their show. Theater goers then pay the producing company for a ticket (usually $5-10--festival badges, to get into the festival, I've seen around $10-20). You then keep a percentage of the house gate (70-80% IIRC) and the Fringe gets their cut.
The producer provides the performance and pays the Festival to perform there and then collects the house; The Festival acts as host/landlord, providing a house to perform in, some publicity, and a small tech crew (someones it's a light/sound guy, maybe two, sometimes a stage manager.)
If you find a producer willing to do your show and you do not act as producer yourself, DO NOT GIVE YOUR SCRIPT AWAY! There are producers out there that will prey on young writers. Have a contract in hand, know what your rights are as a playwright (join the DGA), and make sure YOU GET PAID.