If someone could explain this to me I'd be very grateful, thanks.
The primary reason for the asymmetry of the Ly$\alpha$ line is bulk motion of neutral hydrogen, i.e. accreting gas (causing a blueshifted line) or galactic outflows (causing a redshifted line).
Mechanism of the Ly$\alpha$ double peak
In general, for Ly$\alpha$ photons produced in the center of a blob of neutral hydrogen (i.e. a galaxy), the photons must escape in a double-peaked line. The reason for this is the large cross section $\sigma$ for photons in the line center. The Ly$\alpha$ photon cannot travel very far before it hits a hydrogen atom, is absorbed and re-emitted almost instantly in another direction. But due to the thermal motion of the gas atom, the photon picks up a small Doppler shift at each scattering, either to the blue (if emitted in the direction that the atom is traveling) or to the red (if emitted in the opposite direction). In this way the photons slowly diffuse not only in space but also in frequency. As the move away from the line center $\sigma$ decreases, rendering the gas less optically thick, and thus it becomes easier to escape after having diffused either to the red or the blue side of the line center (just a few Ångström).
Redshift caused by outflows
If gas is outflowing from the galaxy, as is often the case in high-redshift galaxies due the enhanced star formation, then in the reference frame of an outflowing shell of gas, the "blue" photons will be redshifted and thus be in the line center where the opacity is high. Hence they will scatter multiple times and be trapped in this gas, and once again diffuse in frequency. On the other hand, the "red" photons will be even redder in the reference frame of the outflowing gas, and pass through freely. The photons trapped in the shell can escape once they have diffused to the red side.
Thus, outflows have the effect of erasing the blue peak, not by absorbing the photons, but rather by turning them into red photons.
Blueshift caused by (cold) accretion
In case of large masses of gas accreting onto the galaxy, which also should happen in the process of galaxy formation, the story is reversed; in the reference frame of the accreting gas the red photons are seen in the line center, are trapped, and must diffuse to the blue side before they can escape.
Redshift caused by a neutral intergalactic medium
At very high redshifts, say above 5 or so, when we are close to the Epoch of Reionization where the neutral fraction of the intergalactic medium is high, the blue side of the line will also be erased, for the same physical reason as with outflows. Only in this case it is not neutral gas flowing out from the galaxy, but simply the neutral gas in the expanding circumgalactic medium.
If you want to delve deeper into this, I cannot help but advertising a little for my thesis which is exactly on this topic. Chapter 3 discusses the physics of Lyman $\alpha$ resonant scattering.