  Sub-arcsecond localization of the repeating fast radio burst FRB 121102
revealed its coincidence with a dwarf host galaxy and a steady (`quiescent')
non-thermal radio source. We show that the properties of the host galaxy are
consistent with those of long-duration gamma-ray bursts (LGRB) and
hydrogen-poor superluminous supernovae (SLSNe-I). Both LGRBs and SLSNe-I were
previously hypothesized to be powered by the electromagnetic spin-down of
newly-formed, strongly-magnetized neutron stars with millisecond birth rotation
periods (`millisecond magnetars'). This motivates considering a scenario
whereby the repeated bursts from FRB 121102 originate from a young magnetar
remnant embedded within a young hydrogen-poor supernova remnant. Requirements
on the GHz free-free optical depth through the expanding supernova ejecta
(accounting for photo-ionization by the rotationally-powered magnetar nebula),
energetic constraints on the bursts, and constraints on the size of the
quiescent source all point to an age of less than a few decades to a century.
The quiescent radio source can be attributed to synchrotron emission from the
shock interaction between the fast outer layer of the supernova ejecta with the
surrounding wind of the progenitor star, or from deeper within the magnetar
wind nebula. Alternatively, the radio emission could be an orphan afterglow
from an initially off-axis LGRB jet, though this might require the source to be
too young. The young age of the source can be tested by searching for a time
derivative of the dispersion measure and predicted fading of the quiescent
radio source. We propose future tests of the SLSNe-I/LGRB/FRB connection, such
as searches for FRBs from nearby SLSNe-I/LGRB on timescales of decades after
their explosions.
