I understand this means "okay. I got it". But is this phrase actually an impolite way of saying so? Like when if you are annoyed or frustration with someone telling you the same thing over and over.
Yes, your understanding is correct.
알았으니까(요) sounds like saying "Okay, I got it (so let's leave it at that)", not wanting to dwell on the subject anymore. It is often used when you want to dismiss something, like when you're annoyed by what the other person said.
Its base form 알았다 (past tense of 알다, to know) tends to have this implication, because the past tense in this case actually signifies a completion and its resultant state (like present perfect in English), which in this case stresses that the person has understood it and therefore there's no more to talk about it.
If you want to say "I understand" in a more polite way, 알겠습니다 (from 알겠다) is the better choice. 겠 is similar to the modal verbs "could" or "might", but here it gives the sense of "I really seem to grasp it now" in contrast to 알았어's "I already got it". A lot depends on the tone of speech too though.
Finally, the most polite way would be 잘 알겠습니다 or 잘 알겠어요, where 잘 adds the sense of "fully / satisfactorily". Making it a full independent sentence like this instead of wrapping it in the sentence ending -니까 indicates you pay full attention/respect to the given subject, so it carries a lot more weight.