[Spivak]

> Please explain to my mother how this system would work.

Instead of casting your vote by giving it to a single person you give it to many different groups interested in keeping the election fair. These groups can be anyone -- advocacy groups, political parties, independent auditors, or local governments. They all receive your vote, verify that it's correct, and sign their name below your vote on a public ledger which can't be tampered with without everyone knowing.

Hell, you don't even really need a digital blockchain for this kind of system to work.