You don't, but also you don't have to. Voter rolls are cross referenced with other sources of data to verify citizenship. ID is required to submit a non-provisional ballot even during early voting if you're not in your designated precinct.

Also just generally it's a severe federal crime to vote illegally, so people who are here illegally aren't out en masse publicly tying their identity to federal felonies.

Exactly, what you give them to apply is not everything they use to verify you.