My understanding is the process in a lot of these states is that you put the anonymous ballot in an envelope, then put that envelope in an outer envelope and sign the outer envelope. The idea being that the officials check the signature on the outer envelope, if it's valid and non-duplicated they open it and put the now-anonymous envelope with the sealed ballot in the pile of votes to be counted. This is then opened by the vote counter, and this person who has no idea of the origin is the first to see what the contents of it are.

Ooooohhh. That makes sense. I thought they meant actually signing the ballot itself, which is why I was so confused.

You give up an element of this anonymity and repudiablity when using a mail-in ballot system instead of a truly anonymous in-person voting system. The votes are still tallied anonymously, and it's impossible to post-hoc validate who you voted for, so it serves the anonymity purposes since you can't check whether your investment in buying votes was successful.

Every state that does mail-in voting at all requires a signature. Alabama, South Carolina, North Carolina, and Missouri also require a witness -- Missisipi requires a notary.