He likely has an infection/abcess and/or inflammation possibly from a spider bite or other insect.
CALL THE VET BACK, or call your own vet. Ask what sort of improvement you should see right now. If you don't feel comfortable with the answer, go to a different vet.
You SHOULD NOT self-diagnose yourself or your pet. This can be dangerous. There are too many things that look alike and can even be difficult for a vet to determine without specific tests. You always should take the animal to the vet RIGHT AWAY, when there is something going on. This can also save you money in vet bills due to a much easier recovery from whatever is going on.
CALL YOUR VET/CALL YOUR OWN VET, and your own vet is the one who can see to the actual diagnosis and recovery of your puppy. The emergency vet is there for you, when your own vet cannot or is not open. Call your own vet and get the puppy in for an appointment. In the least, call your own vet and tell them what is going on, what happened and ask what they think at this point in time. If you think the swelling is still extreme right now, you need to tell them that. Sometimes they may think it is just a little swelling and not too bad. You need to describe exactly what is going on.
I, personally, would not wait in a situation where there is a lot of swelling. This can easily involve the eye, and simply is not a good thing either way. You can try to place a warm compress on the swelling, but this does not replace a consult with the vet so your puppy does not end up losing his eye, or vision in that eye.