Those are petechiae. It is caused by tiny broken capillaries. It appears you might have scratched the spider bite and may have broken the capillaries. If you will notice, the petechiae are in the shape of elongated scratches.
I had a similar spider bite on my forearm which formed a blister eventually, and it did not stop itching until I removed the fluids from the blister and caused it to bleed, and then disinfected the whole thing.
