Your friend didn't understand what it meant because it's not idiomatic English, it's not natural native-speaker English, and it's flat-out wrong. It has to be something like "Is it getting late for you?" or "Isn't it getting late for you to be up and about instead of preparing for bed?" The semantics of those two sentences are quite different from the semantics of the original.
It's nonsense to claim that the sentence "Are you getting late?" is grammatical, simply because it doesn't work semantically. We ignore the ungrammaticality of idiomatic solecisms simply because they are semantically clear: we know what they mean because we use them all the time, and meaning is more important than grammar. We can safely ignore the grammatical status of a piece of semantic nonsense, because whether it's grammatical is irrelevant, unless Noam Chomsky's trying to make a linguistic point about colorless green ideas sleeping furiously.
To claim that "Are you getting late?" may refer to someone about to be late to an appointment or to some woman whose menstrual period hasn't begun at the normal and expected time is stretching the language beyond the breaking point, I'm afraid. Although we may say that pregnancy, lateness, uniqueness, and death are technically ungradable but de facto gradable (because we do grade them when we speak), the fact is that there's a clear line, in this case, between being late and not late: 9 p.m. is the listener's normal bedtime.