Recent polarization measurements of fast radio bursts (FRBs) provide new
insights on these enigmatic sources. We show that the nearly 100% linear
polarization and small variation of the polarization position angles (PAs) of
multiple bursts from the same source suggest that the radiation is produced
near the surface of a strongly magnetized neutron star. As the emitted
radiation travels through the magnetosphere, the electric vector of the X-mode
wave adiabatically rotates and stays perpendicular to the local magnetic field
direction. The PA freezes at a radius where the plasma density becomes too
small to be able to turn the electric vector. At the freeze-out radius, the
electric field is perpendicular to the magnetic dipole moment of the neutron
star projected in the plane of the sky, independent of the radiation mechanism
or the orientation of the magnetic field in the emission region. We discuss a
number of predictions of the model. The variation of PAs from repeating FRBs
should follow the rotational period of the underlying neutron star (but the
burst occurrence may not be periodic). Measuring this period will provide
crucial support for the neutron star nature of the progenitors of FRBs. For FRB
121102, the small range of PA variation means that the magnetic inclination
angle is less than about 20 degrees and that the observer's line of sight is
outside the magnetic inclination cone. Other repeating FRBs may have a
different range of PA variation from that of FRB 121102, depending on the
magnetic inclination and the observer's viewing angle.