HQ is a building with highly paid bureaucrats sitting in Washington, DC or its suburbs. A line office would be say a branch of US Immigration and Customs that you deal with at your local airport.

I'm not going to lie; we're all bureaucrats with greater or lesser degrees of pay, although I think we top out at 160k, so by Silicon Valley standards, highly paid is debatable...

Line offices are at the pointy end of the stick where you actually deal with citizens. In my particular case we do a lot of permit issuance and resource management, but yeah think local branch office where you go when you want something.