There are several sources of the asymmetry in the Standard model. One is just from the strong interactions (QCD). Understanding this requires knowing about interference phenomena, It is easier to explain for the simpler but analogous case of QED (quantum electrodynamics). So consider e+ e-  → μ+ μ- collisions. The forward direction is that of the e-, backwards is that of e+. Once in a while a photon (γ) is emitted from either the e+ or e- before the collision (“initial state radiation”) or from the μ+ or μ- after the collision (“final state radiation”).  So we are looking at the process e+ e- → μ+μ-γ which, by the way, is counted together with e+ e- → μ+μ- in the measurement of the asymmetry. Now, recall from your basic electromagnetism that radiation is preferentially emitted in the direction of the charge that emits it. So a photon from the e- is emitted in the forward direction (and, likewise, a photon from the e+ is emitted backwards). 

Next compare with final state radiation. Since we want to consider interference effects we need to have a configuration that includes a photon emitted into the forward direction to interfere with the initial state radiation from the e- (or a backwards γ to interfere with the e+ initial radiation).  Suppose the collision results in a μ- in the forward direction. Then if it emits it will do so in the forward direction. Likewise for a μ+. So both of these can interfere with the e- initiated initial state radiation in the forward direction. But having opposite charge the μ+ and μ- give opposite interference: μ+ interferes constructively while μ- interferes destructively. This increases the probability of μ+ going forward and μ- backwards.  Similarly, for e+ initiated initial state radiation (backwards) the constructive interference is for the oppositely charged μ- (moving backwards to radiate in the backwards direction). So we see that in both cases we obtain a negative FB asymmetry from this process. 

The case of quarks emitting gluons is technically a bit more involved, but qualitatively parallels exactly the discussion above.