The alien parasite rewrites people's DNA, and also (unstated in the episode because the writers thought DNA was magic) forced some sort of redevelopment of the body so that the DNA changes would actually affect the body's morphology. I'm not entirely sure how that would work, because that's the kind of thing that involves stem cells and uterine environments and all kinds of chemical signalling, but it apparently happened, and Geordi's eyes would have redeveloped as part of that.
CRUSHER: There, in the thymus. Some sort of parasite. It's using Susanna's immune system to spread genetic instructions.
OGAWA: How did it get there?
CRUSHER: She could have become infected during the original mission. It's small enough to have entered through any of the mucous membranes. But it certainly isn't behaving like a typical parasite. It isn't feeding off of her. It's transforming her DNA to match its own.
OGAWA: There's not much of her original DNA left.
CRUSHER: And we're going to need unaltered genes or we'll never get her back. We've got to get that thing out of her now.
(The Next Generation Transcripts - Identity Crisis)
Geordi was born blind due to a birth defect (Star Trek TNG Bible, page 7; scanned PDF).
If the birth defect was not caused by a a genetic defect, then the real question is not why he's no longer blind when he's transformed into another creature, it's why he's blind again when he reverts back to being human. And to that I'm sure there's a perfectly reasonable— look behind you, it's a reset button!
If, instead, it's caused by a genetic defect, the re-re-development of Geordi's eyes would result in him being blind again when he's transformed back.
The temporary regeneration of Geordi's eyes in Star Trek: Insurrection only muddies the waters. On the one hand, it suggests that the defect isn't genetic and that his eyes have finally been fixed. On the other hand, clearly something strange is going on, since he's back to having cybernetic eyes in Star Trek: Nemesis.
Either way, the alien creature he was turning into didn't have that birth defect, so his eyes worked when he was the creature.