An example- go to the poll. Cast a vote and get a print out with a private key.
Go home and check the blockchain to make sure your vote is recorded. If your vote is missing, report somewhere official with your sheet of paper.
There’s lots of variations, but anonymity is only given up if your vote is missing. Then you just vote again. Smart contracts make sure your original lost vote can’t be applied.
The US needs something like this because currently our digital voting leave no record and there’s no way for an individual to confirm their vote was counted accurately.