  Lyman-alpha (Ly{\alpha}) photons from ionizing sources and cooling radiation
undergo a complex resonant scattering process that generates unique spectral
signatures in high-redshift galaxies. We present a detailed Ly{\alpha}
radiative transfer study of a cosmological zoom-in simulation from the Feedback
In Realistic Environments (FIRE) project. We focus on the time, spatial, and
angular properties of the Ly{\alpha} emission over a redshift range of z = 5-7,
after escaping the galaxy and being transmitted through the intergalactic
medium (IGM). Over this epoch, our target galaxy has an average stellar mass of
$M_{\rm star} \approx 5 \times 10^8 {\rm M}_\odot$. We find that many of the
interesting features of the Ly{\alpha} line can be understood in terms of the
galaxy's star formation history. The time variability, spatial morphology, and
anisotropy of Ly{\alpha} properties are consistent with current observations.
For example, the rest frame equivalent width has a ${\rm EW}_{{\rm Ly}\alpha,0}
> 20 {\rm \AA}$ duty cycle of 62% with a non-negligible number of sightlines
with $> 100 {\rm \AA}$, associated with outflowing regions of a starburst with
greater coincident UV continuum absorption, as these conditions generate
redder, narrower (or single peaked) line profiles. The lowest equivalent widths
correspond to cosmological filaments, which have little impact on UV continuum
photons but efficiently trap Ly{\alpha} and produce bluer, broader lines with
less transmission through the IGM. We also show that in dense self-shielding,
low-metallicity filaments and satellites Ly{\alpha} radiation pressure can be
dynamically important. Finally, despite a significant reduction in surface
brightness with increasing redshift, Ly{\alpha} detections and spectroscopy of
high-$z$ galaxies with the upcoming James Webb Space Telescope is feasible.
