What is the significance of tire minimum pressure? Its quite intuitive that tires have maximum pressure figure - inflate it more and something could explode. Furthermore, it is common knowledge that under-inflated tires result in snake bites.

What puzzles me is that all tires indicate minimum pressure. 

This could be somewhat justified about DH tires, which tend to be produced with thick sidewalls and thick soft thread pattern, resulting in reduced  pinch flats (among other advantages). However, on different cross-country tires I have seen minimum pressures from below 2bar to 2.7bar at least. Is this just marketing or does the number mean anything? 

Furthermore, how can the manufacturer know what the riding terrain and rider's style is? In my opinion those affect chosen tire pressure a lot more than sidewall thickness.
The minimum pressure is an estimate that will most probably work for the majority of people in generic riding conditions. Not going below the minimum pressure may prevent:


loss of control due to insufficient traction
pinch flats
burped air in case of tubeless
tyre unseating
rim bending on harsh hits
tyre carcass premature wear
rotation of the tyre/tube combo within the rim (can cause tube valve failure)


Of course, that number just an estimate. An aggressive rider, a heavy rider, or a really rough terrain may require higher than the minimum suggested values.