Normally ごめんなさい is translated as sorry or excuse me, but that seems like a translation for non-native speakers. In the same way, しつれいします can also be translated as excuse me or pardon me. But it really means "I'm being rude". So... what does ごめん really mean?
めん(免) in ごめん(御免) means 'forgiveness/to forgive', like in the verb '免ずる/免じる' (which I think is the literary or archaic form of '許す'). なさい makes it imperative, so ご免なさい literally means 'Please forgive (me)', like 免じてください/許してください.