It's largely for primary voting in states that have partisan primaries. If you're registered as a Democrat then you receive a ballot with only the Democratic candidates.
Of course one could argue that all primaries should be non-partisan. Or that governments shouldn't spend money on running partisan primaries on behalf of political parties which are private organizations.
I understand having to select a party for primaries & the pros & cons of it. That is a different & even more complicated issue.
I just don't understand why which party I voted for needs to be "public" record. I don't believe that companies & the general public need to know. I've used campaign software that shows how incredibly easy it is to select a list of people & market towards those who voted one way or another in the primary.
Of course one could argue that all primaries should be non-partisan. Or that governments shouldn't spend money on running partisan primaries on behalf of political parties which are private organizations.