It has larger grains. It is called "kosher" salt because larger grain salt (typically coarser than even typical kosher salt) is used to drain the blood from meat and fowl, a necessary step in rendering it kosher by Jewish law.
It's a larger crystal, which makes it easier for the salt to pull water out of the watermelon, versus (I'm guessing) just making its way into the watermelon as a dissolved solid.
My understanding for the mechanism by which the salt makes the fruit sweeter is by increasing the concentration of sugars in the flesh that's now surrounded by salted water.
Kosher salt is named for the application of drawing water from meat (which is itself part of the kosher preparation of meat) so it's not too surprising that the salt is also well suited for drawing water from fruits like watermelon.
I use table salt on watermelon and it's great too. I can believe that kosher salt would be better, but I doubt most people would consider the difference immense. But palates vary, so, if you've got both salts around might as well use the bigger crystal.
But also it doesn't have additives like table salt (e.g. iodine and anti-caking agents). Iodine at least has a definite flavor.
It's also very consistently made (though not between manufacturers) unlike some sea salts, and relatively inexpensive so it's the go-to kitchen salt (as against finishing salt) for a lot of cooks; you know what you get in a pinch every time.