Someone raised in the Washington D.C. metropolitan area, by parents who moved to the area to work in the government or a related field (law, lobbying, defense contractors, etc). The strictest definition would only apply to people living within the Capital Beltway, but more usually it refers to any county the Beltway passes through or encompasses (Washington D.C., Arlington, Alexandria, Falls Church, Fairfax, Montgomery, and Prince George's County). As military brats are likely to know an unusual amount about the military, so beltway brats are likely to know an unusual amount about politics and government.
Alice: Charlie kept talking to the teacher about tort reform. How does he know so much about politics?
Bob: Oh, he grew up here in D.C., and his mom works for the Department of Agriculture-- he's a total beltway brat.