Judge Judy used to be a sitting judge but she isn't any more. When they say those are real cases, they mean they go through a real small claims court docket, looking for cases. Then the folks on either side agree to withdraw their civil complaints in exchange for being on a TV show. But they can't appeal, because it's not a real courtroom, and she's not acting as a real judge. Same goes for Judge Joe Brown. It's a mock trial, a moot court, for entertainment purposes only.
It is more a legal arbitration. To the best of my knowledge, private arbitration is a binding action in lieu of using civil courts. I'm not entirely sure what the pros and cons are other than ability to enforce actions.
The main thing for Judge Judy et al is that both parties are paid for their appearance and any settlement is taken out of the loser's payment and added to the winners. They sign a binding arbitration agreement which is quite common in civil cases. The TV judge is acting as the arbitrator and, like any arbitration proceeding, the decision is final and cannot be appealed.
The way it works is that there is a set fund for each case. The amount awarded to the wining party is taken out of that fund (I believe if the award is more than the fund then they just get the amount in the fund). Any amount left over is split evenly between the two people. The losing person may leave with nothing, but they don't leave owing anything. It actually has advantages for both parties. If you lose you aren't out anything, if you win you are guaranteed some money...you aren't left to try to collect from the loser.
From what I understand by a person who was on one of these shows, the staff checks out filings in petty claims court on a regular basis. If it looks like something that could be bizarre, off-beat, or involving nutsy-cuckoos, both parties are approached about being on the show. There is (or was) a $5000 cash award for each case. If the parties agree to appear, the judgement comes out of this cash award and the remainder is split between the two.