[MPaulHolmes]

You want to keep the inductance as small as possible, so the capacitors, mosfets and diodes need to be really close. If you had no caps, the pulses would have to come from the batteries, which would be far away, and the ESR would be high. The large inductance would cause very large voltage spikes as the current tried to change quickly, which would probably destroy the mosfets. The diodes are just there to take the current that's flowing through the motor windings when the mosfets are off. They don't really smooth anything out. If the diodes weren't there, the current would piledrive right though the mosfets from drain to source, since current can't stop instantly. It's like a water pipe with current racing through it, and all of the sudden, you close a valve. The pipe will probably explode since the water will still want to keep going. The diodes give an alternative path for the water to go.

EDIT: I feel like most of what I said above makes sense to me, but let's see. a sudden near short circuit (closing the mosfets) makes an almost short circuit. Tons of current wants to suddenly flow. With capacitors charged, it can mostly come from them, which is a very short distance to the mosfets.

What causes the oscillations in voltage seen by the mosfets from large distances? Something about LR or LC or LRC or who knows what. Could someone explain that?