Yes, “밥 먹었어요?” is translated as “Have you eaten?” and yes, it is used as a greeting. I can see how it can be confusing to those learning the language.

There’s actually a heartwarming meaning and history as to why Koreans use it as a greeting. Food was scarce after the war in the 60s and 70s, and Koreans would ask each other if they’d eaten anything to make sure they were okay. Basically, asking if you’ve eaten is asking how you’ve been.

These days it’s a greeting out of habit, sometimes they’re really asking if you’ve eaten.

So, how do you say “How are you?” in Korean?

There are different ways.
Here are the most common ones in a formal tone.
1. 잘 지냈어요? – How are you?
2. 잘 있었어요? – Have you been well?
3. 어떻게 지냈어요? – How have you been?