It's so you can participate in primary elections for that party. Some people who vote Dem in the general election will sometimes register as Republican so that they can vote for spoiler candidates in the Republican primary. There's not enough people doing it for that plan to really have any impact though.

But the same logic applies if you are trying to hide your votes from nosy family. Family members have enough info on you to be able to look up your party registration so if you are afraid of them causing drama, you can register as whatever party the rest of your family is registered as and then vote for the other side in the general election. The actual votes you cast cannot be traced back to you... Which can raise other concerns like not being able to verify if your vote actually counted.