Air flows spanwise because of the way the upper surface pressure distribution behaves on a swept wing. Refer to the diagram:
At each section of the wing, the pressure distribution along the chord looks like the upper diagram. The front of the upper surface has a favourable pressure gradient - that is, the local pressure drops as one moves aftwards.
This means that there's no real obstacle to the boundary layer moving aftwards - in fact, the pressure distribution encourages this.
But, after the suction peak is passed, the pressure distribution is adverse for aftwards motion - the local pressure is increasing as the air moves aft. The boundary layer is also thickening, so there's less energy to overcome the adverse gradient.
On a straight wing, the pressure distribution is pretty much the same spanwise, so the air "has no option" but to continue trying to move aft; a trailing edge stall/separation will occur if it just can't overcome the pressure gradient.
But, on a swept wing, there's an option - spanwise. On the lower diagram of a swept planform I've drawn in two lines, roughly corresponding to the 1.0 Cp point on the upper chart. The arrows indicate the adverse gradient. If our air molecule has reached the aft red line, and has 'run out of energy' to move aft, it can move outwards instead - where the pressure gradient is not adverse, since moving spanwise actually means moving forward
relative to the local chord.