>How does that part work in practice? Are these forms something someone from a poor uneducated background can actually realistically fill out?

Yes, at least in the 3 states where I know anything about how the systems work. It's all the same information you fill out to get your driver's license, kid's school lunch which, contrary to the prevailing trope around here, being poor does not make you too stupid to do. It's way less arcane than reporting loan interest on your taxes or trying to register a used car last registered by a dead guy.