The practice of making public whether someone has voted dates back a very long time. There were practical reasons for it when it originated, and many of those remain valid today.
Originally, everyone in town knew most everyone else and could see them physically walking into the polling place. So long before electronic records were kept, the question of who voted was public information.
Even today, I run my local polling station and I know and recognize a significant portion of the voters in my precinct.
The fact that the list of who votes is made public protects against several kinds of abuses. It goes a long way to protect against "dead people voting" and other kinds of ballot box stuffing if the list of voters is public so anyone can review it and potentially catch such abuses.
The value of transparency goes above and beyond protecting against abuses (for which there are other methods) - it also provides assurance to the public that there is protection against those abuses, and that those abuses are not taking place.
i.e. 'Justice must not only be done, but must also be seen to be done'