The growing worms absorb nearly all nutrition the hosts take in, leaving them just enough to survive. Then they start eating the internal organs leaving only the nervous system and locomotive muscles. Then they release chemicals that cause the host to drag itself into a small body of water. The worm exits the body and starts the life cycle anew.
It's likely the reason the person saw the spider walking around the house was because the worm had matured to the point it had plugged itself into the dying spider's brain and was piloting it towards water.