Given the situation, being interrupted by someone which is inherently rude-ish, saying 'hear me out' is probably one of the most polite ways to reassert yourself in the conversation. Other options, such as the following, are a slightly stronger response as they're more imperative: 'Let me finish.' 'Let me speak.' 'Stop interrupting me.' It'll also depend on your tone, body language, etc, but 'hear me out' is a safe option. 'May I finish, please?' Is also good...or adding 'please' to the request helps soften it.

The meaning of your sentence is ambiguous. You probably mean … if someone interrupts me while I’m talking. But could mean … if someone interrupts me while I’m giving a talk. In neither case is ‘Hear me out.” a natural thing to say, but it isn’t rude. ‘Hear me out’ is something you might say before starting to give a long explanation. It doesn’t address the fact that you were interrupted. For example: It’s a complicated problem but I think I understand it and can explain it to you. Hear me out.

The magic word is "please". It transforms commands (rude) into requests (polite): "please hear me out" "please let me make my point" "please, may I finish".