My only guess is that everyone is homeschooled or people are educated via the various temples that exist in most towns and cities.
It would be reasonable to believe the prominent temples and religions of Tamriel would want the populace to be able to read and spread their religion to others.
Eg. The Imperial cult in Morrowind liked to talk about missionaries and conversions. Ensuring your converts could read your holy books would make religious conversion easier.
Of course that doesn't explain literacy amongst religions which don't practice conversion with outsiders such as Argonians and Hist worship. There would have to be multiple means by which people in Tamriel are learning how to read and write.
The best answer would be a form of public schools in every provence, but I don't think we really see any?