What is the significance of tire minimum pressure?
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.