Why do Koreans ask my age almost immediately after meeting me?

Because Korean grammar can't proceed without the answer. It isn't nosiness — it's a genuine structural requirement.

Korean has multiple speech levels, and which one you use depends on the relative age and status between you and the person you're talking to. There are also different words for the same relationship: an older brother is 형 if you're male and 오빠 if you're female. Until someone knows whether you're older or younger, they literally can't choose verb endings without risking rudeness in one direction or excessive stiffness in the other.

So the question arrives early, often within the first minute, and it's usually followed by relief once it's settled. If it helps: they're not judging your age, they're looking for the right grammar. Answering with your birth year ("I'm a '95") is the most natural way to respond — Koreans often use birth year rather than the number itself.