Eating watermelon with salt is to enhance the sweetness of the watermelon.
Well it's not really "enhancing" but with the saltiness of the salt, it brings out the sweetness in the watermelon.
You can try eating plan watermelon, see how it taste like, and then sprinkle it with just a little bit of salt and try the difference!
It's the same concept when you try to make soup or savory stew, sometimes I add a little sugar to give it more depth.