Bleach does kill mold spores, but even after they're dead they can still cause an allergic reaction. You can wipe a surface clean, but if you're dealing with a porous material, like wood or grout, getting rid of all the deeply ingrained spores can be tricky. Worse yet, bleach strips the spores' color, making it difficult to see what's still there.
The real remedy is to remove mold completely—with a detergent and water, or a scrub brush and a HEPA-filter vacuum. Then, fix the root of the moisture problem so mold doesn't regrow.