As the description states, this achievement is for destroying 100 Anthills or spider nests. These come in the form of large brown mounds coming out of the ground almost like a volcano and you will see multiple ants or spiders crawling out from the top.
Just to clear things up, blowing up the hills with pesticide weapons does NOT count towards the achievement. So this means that you have to go up and plant the charge yourself to destroy the ant hill. To make this easier, you can run up and unload an assault rifle in to the hill and it will eventually close off giving you a brief window to plant the charge, though in most cases you can plant the charge easily without having to do this.
A quick warning, if you are playing with bots be careful because they will unload on you if you are planting the bomb and they want to close the hill first. So I would advise circling around to the opposite side of them to save yourself frustration.
If you are only wanting to get the achievement quick, you can load up ditch slap and plant on the hill that is in the beginning and advance to the next point with 3 more ant hills behind the wall you have to blow up. After blowing them up, then restart the stage. You won't earn credits this way but your hills destroyed and kills will all count in the stats.