Here's the short answer-

They get paid to do it.

Most "Trash TV" shows (Maury, Springer, Sally-Jessy, Cheaters, most TV courtroom shows) actually PAY their guests to make fools of themselves on national TV. In some of the shows, they pay pretty well, too. Not only that, but the more lurid or scandalous the story or the more violent the confrontations, the more money the guest will likely get.

And since most of the guests of shows like Jerry Springer are missing a number of teeth, chances are that they'll jump at the chance at being on television... even if it IS to make themselves look like fools.

In the case of TV courtroom shows, they'll actually pay BOTH litigating parties and will even pay for the damages stemming from the originating conflict. So- in essence- they're reduced to doing what the often-toothless do on shows like Springer- look like dolts on TV.