  We investigate how a fast radio burst (FRB) emitted near a magnetar would
propagate through its surrounding dipole magnetosphere at radii $r=10^7$-$10^9$
cm. First, we show that a GHz burst emitted in the O-mode with luminosity $L\gg
10^{40}$ erg/s is immediately damped for all propagation directions except a
narrow cone along the magnetic axis. Then we examine bursts in the X-mode. GHz
waves propagating near the magnetic equator behave as magnetohydrodynamic (MHD)
waves if they have $L\gg 10^{40}$ erg/s. The waves develop plasma shocks in
each oscillation and dissipate at $r\sim 3 \times 10^8 L_{42}^{-1/4}$ cm. Waves
with lower $L$ or propagation directions closer to the magnetic axis do not
obey MHD. Instead, they interact with individual particles and require a
kinetic description. The kinetic interaction quickly accelerates particles to
Lorentz factors $10^4$-$10^5$ at the expense of the wave energy, which again
results in strong damping of the wave. In either propagation regime, MHD or
kinetic, the dipole magnetosphere surrounding the FRB source acts as a pillow
absorbing the radio burst and re-radiating the absorbed energy in X-rays. These
results constrain the origin of observed FRBs. We argue that the observed FRBs
avoid damping because they are emitted by relativistic outflows from
magnetospheric explosions, so that the GHz waves do not need to propagate
through the outer equilibrium magnetosphere surrounding the magnetar.
