We don't know how it works in detail but we can see which parts of the brain are doing stuff at any given time, and we've had a lot of progress associating different functions with different areas of the brain.
http://en.wikipedia.org/wiki/Functional_magnetic_resonance_i...
No, we can detect changes in blood flow, and we've posited that there's a correlation between this blood flow and brain activity, and found that that is consistent with the way the blood flows alter.
This does not in any way rule out that the parts of the brain that are not showing up in the fMRI are participating in the thinking process at that moment. We have no idea whatsoever of how the thinking process itself unfolds.
