This is my pet spider beside its molted exoskeleton. Notice that the spider is bigger than the exoskeleton it came out of. How is that possible?
Like other arthropods, spiders grow a new, larger exoskeleton inside the old one. The new exoskeleton is flexible, so it fits inside the old exoskeleton even though it is larger. You can read more about the process in Thoughts on an Exoskeleton.