Professor Toad is not the result of some magical accident or dreadful curse. He is, in fact, a toad and proud of it. He teaches the apprentices magic in the meanest and most unfun way he can. Most apprentices would rather skip his lectures, even Pip (the little green apprentice).
Professor Toad grew up in a humble swamp and never really fit in with the other tadpoles. His family disapproved of his pursuits in magic and haven't been in contact with him since he left. Professor Toad thinks back fondly of the swamp but has very bitter feelings towards his friends and family. That said, he will always be proud of his toad heritage.