This has nothing to do with "Swamp People," though I suppose you could use one of these saws to cut up a gator.
If you look at these tools from Bosch and DeWalt, you can see why they're called what they are. But it's kind of funny that they sell them in Europe (where there are no gators) but don't sell them here (where if the History Channel is to be believed – we're being overrun by them). Saws of this type have been marketed in the U.S. but never caught on. In Europe they're used like recip saws and to cut particular types of masonry. Here's a video (at the 1:35 mark) that shows an Alligator saw being used on Porotherm block.
FWIW, I think Wen sold one in the US at one time- I saw a butcher using it.
Now that you mention it - I remember seeing this type of saw with a Wen label on it (back when the brand color was red; now it's blue).