My pet was seen 3 months ago for an eye infection and I got some ointment that made it better. The infection is back and I don't understand why I have to come back in with my pet to see the veterinarian.

This is a common problem we see in all facets of the health professions.  A client with a pet that had an ailment ( an eye infection for example) several months ago, came in for an examination and was prescribed a topical ointment.  The owner believes the pet has the same problem and wants a refill.
First, legally, we cannot dispense medication without seeing the patient. It is a violation of our state laws.
Second, perhaps there is more than just a recurrent infection in the eye. Let's pretend the first eye problem was a simple infection and we made sure there was no scratch to the eye. Suppose the first medication had a steroid in it to reduce irritation.  If this same medicine was prescribed to a patient with a small scratch (corneal ulcer), the events would be disastrous.  Steroids would cause the corneal ulcer to worsen, possibly resulting in the loss of the pet's eye.