This achievement unlocks when you kill 8 of the large purple spiders within 40 seconds.
Or at least it should unlock that way. It unlocked for me around the time I killed my eighth spider on the moon, but I took much longer than 40 seconds to kill them. CanadianWhizKid confirms that it unlocked for him under similar conditions, so as of right now (December 5th, 2017) the achievement unlocks when you kill 8 spiders total, regardless of time spent doing it.
If for some reason that doesn't work for you, here's the actual solution:
Discover the spiders using the Discovery Tool. Just like all the other things on the moon, the spiders are very common.
Now you need to create an enclosed area. Use the Landscape Tool to dig a large square into the ground, deep enough that they can't escape. Then place 8 of the large spiders inside the hole. Jump inside and murder them. Achievement Unlocked.