With the Multi Unit Spectroscopic Explorer (MUSE), it is now possible to detect spatially extended Lyman α (Lyα) emission from individual faint (M UV ∼ −18) galaxies at redshifts, 3 < z < 6, tracing gas out to circum-galactic scales comparable to the dark matter halo virial radius. To explore the implications of such observations, we present a cosmological radiation hydrodynamics simulation of a single galaxy, chosen to be typical of the Lyα-emitting galaxies detected by MUSE in deep fields. We use this simulation to study the origin and dynamics of the high-redshift circum-galactic medium (CGM). We find that the majority of the mass in the diffuse CGM is comprised of material infalling for the first time towards the halo center, but with the inner CGM also containing a comparable amount of mass that has moved past first-pericentric passage, and is in the process of settling into a rotationally supported configuration. Making the connection to Lyα emission, we find that the observed extended surface brightness profile is due to a combination of three components: scattering of galactic Lyα emission in the CGM, in-situ emission of CGM gas (mostly infalling), and Lyα emission from small satellite galaxies. The weight of these contributions vary with distance from the galaxy such that (1) scattering dominates the inner regions (r < 7 kpc), at surface brightness larger than a few 10 −19 cgs, (2) all components contribute equally around r ∼ 10 kpc (or SB∼ 10 −19), and (3) the contribution of small satellite galaxies takes over at large distances (or SB∼ 10 −20). Compared to stacked MUSE observations, we show that we can reproduce the observed Lyα surface brightness profile closely, to within at least 0.2 dex, spanning over two orders of magnitude in surface brightness, and out to beyond the halo virial radius. We find that the CGM produces on average slightly blue-shifted (and single peaked) Lyα spectra, reflecting the net inflow of dense circum-galactic gas. Our simulation fails to reproduce the characteristic observed Lyα spectral morphology that is red-shifted with respect to the systemic velocity, with the implication that the simulation is missing an important component of neutral outflowing gas.