Winds can be strong on Mt. Washington for many reasons, but this area is known for its weather. It is incredibly erratic and can’t make up its mind. The reason for that is because, as Wikipedia puts it, it is the convergence area for multiple storm tracks. That just means that multiple areas known to be stormy all mix right there.
As for the winds specifically, the previous answer notes, low pressure systems move in and constantly bombard Mt. Washington. This creates interesting wind speeds, and Mt. Washington actually holds the record for highest winds recorded without a cyclone or tropical storm being involved: 231 miles an hour on April 12, 1934. That’s a lot of wind for a mountain.
This is actually a complex situation with a cause that is related to the movement of the jet streams high up in the atmosphere and the location of Mount Washington. Every three days there is a change of wind movement as the low pressure hits the high at the north eastern corner of New Hampshire. This causes weather changes and Mount Washington is just at the spot where, up high, the action is happening.
Despite being in a temperate zone, and despite not being the highest peak in the Appalachians, Mount Washington has subzero temperatures all of the winter, sometimes as low as minus 50 degrees, matching Everest. Then, catching these weather changes, the wind speed lowers the temperature further. It's an unenviable place to visit, yet thousands do. Well somewhere had to gain the reputation of the world's worst weather, and Mount Washington has won it.