Why would this h-bridge driver circuit include capacitor across MOSFET's?

I see you've already accepted an answer but that answer misses an important point about the power feed and the load. For instance, if the power feed is from a voltage regulator then energy transferred back to the power rails when switching an inductive load or a load that can produce energy (a motor for instance), will try and raise the "regulated" supply voltage. This happens quite easily with a series voltage regulator because there is no mechanism inside it to "shunt" away excess energy.

This means that a capacitor (of decent value) is required to be across the voltage rails to "soak" up the excess energy being returned by the load through the MOSFET parasitic diodes. This isn't needed with resistive loads but inductive loads (or motors that can generate when being mechanically rotated) will (or can) force the supply voltage higher than the supposed regulated value.

Path of current flow when a motor is connected as a load and allowed to free-wheel: -

You should be able to see that without a capacitor, there is nothing to prevent the voltage rails rising to an unacceptable level.

It seems to be bypass capacitor to stabilise the power rails

This application note gives the same schematic

the appropriate size will depend on the power rails, the value depends on the noise frequency