To figure this process out, Keizo Takasuka of Kobe University's agriculture science graduate school looked at the Cyclosa argenteoalba spider's web construction. The spider builds two kinds of webs: an orb web, for when the spider hunts, and a resting web, for when the spider is about to molt. The orb webs are sticky, designed to catch flying insects. But the resting webs have way fewer radii and feature fluffy decorative elements:
The zombie enslavement begins when a wasp lays eggs on the back of a spider. The resulting larva feeds off of the living spider's body fluid. About 10 days after hatching, the larva essentially turns the spider into a zombie that constructs a tough cocoon web to protect the baby wasp as it matures.
These cocoon webs were found to be similar to the resting webs but stronger; the periphery of the cocoon web is three times stronger and the center is 30 times stronger. They also had decorative elements that reflect light, presumably to deter flying creatures from getting caught up in the web.
Takasuka examined Cyclosa argenteoalba spiders he collected from shrines in Tamba and Sasayama, and saw that over a 10-hour period, the zombie spiders actually broke apart their orb webs to build a cocoon web instead.
But the spider isn't done once the cocoon web is complete; the larva then direct the spider back to the web's center and kills it, then weaves the web into a tough cocoon for itself. (Watch that chilling video here, if you so dare.)
The entire process is kind of a heart-breaking thing to witness, to see the spider painstakingly break down its old web to construct this cocoon for the parasitic wasp:
The study authors describe the spider as "a drugged navvy." They also write that their findings suggest "the cocoon web in this system has roots in the innate resting web" and the wasps may be controlling the spiders through the hormones responsible for molting.
So, look, if nothing else: just be happy you're not one of these spiders.