Those are fleas and you use the net to catch them off them. It's actually annoying after the first time, especially when it's on someone I like. I just have to stop and help them.
They are fleas and you just hit the villagers with the net to catch the bugs and you will get a flea and a thank you from the villagers. And when you talk to them while they have a flea they might say stuff like I-T-C-H-Y!
They get fleas a whole lot more in New Leaf than ever before it seams. I have a theory that this is a metaphor for the decline of human society. But that's just me.