I don't understand. How is this type of law legal at all under the first amendment? Like, is the answer "just because?" Seems like pure political speech.

In Texas, it’s a fallout from no phones in a polling location

The argument is that the law doesn't limit speech, because you can still tell people who you voted for. What it prevents, is anyone checking on that statement, and using that to coerce you. A secret ballot means you can tell your boss, or husband, or anyone that you voted the way they want.

I think that the usual rational is that allowing it enables people being rewarded or punished based on their votes. In particular, prohibiting it prevents people being pressured by spouses or employers to prove they voted the "right" way.