Low pressure systems have divergence (spreading out) of air at the top of the atmosphere (tropopause) and convergence of air at the surface. Since the air is converging at the surface, it has no other place to go but upwards. This causes the lifting mechanism and hence rising parcels of air, increasing instability.
The way in which low pressure systems are situated is that they rotate in a counter-clockwise direction due to something called the Coriolis Force, which is basically caused by the rotation of the Earth. Since lows do spin counter-clockwise, this rotation pulls up warmer air from the South and colder air behind the center of the low to the North. So what happens is that a cold front that stretches towards the south develops in cyclogenesis while a warm front is created towards the East of the low.
The cold front will basically rise over the warmer air enhancing instability and causing more upward lift, which is why you generally see thunderstorms before a cold front.
As opposed to the low pressure system, the high pressure system has divergence at the surface, so upward lift isn't possible. You need that lifting mechanism to create precipitation. Also, highs spin clockwise.