When you do something airily, you do it in a casual, offhanded way. You might hide your excitement at meeting a movie star by airily saying, "Oh, hi."
If you speak or behave airily, you give the impression that you don't really care. If a local politician responds airily to a citizen's legitimate concerns about crime, she's likely to suffer some criticism for not taking her constituents seriously enough. Airily comes from airy, originally defined as "with ostentatious air."