Electronic – 4 P-FET HBridge

I'm hardly an expert on the subject, but the idea seems to be to better approximate a sine wave by having a period of 0V between the peaks. The explanation is in the document you cite:

The major problem with the push-pull approach [they are talking about a straight-up square wave here] is that the current in the transformer has to suddenly reverse directions. This would be like shifting your car into reverse at fifty miles per hour. This causes a large reduction in efficiency as well as potential for large transients, thus degrading the waveform.

Then they introduce the "Modified Square Wave":

The addition of an extra winding in the transformer along with a few other parts allows output of a Modified Square Wave (often referred to as a modified sine wave by marketing types) while still utilizing a push-pull topology (Figure 5).

The switching cycle is identical to that described in the section on square wave inverters, except for one additional step. In the switching cycle, another step is added which “clears” out the transformer reducing the problems associated with the sudden change in current direction. This is accomplished by the off time shorting winding shown in figure 6.

Up to this point they are only talking about a push-pull topology. The section you ask about is then describing the same technique, but with an H-bridge topology.

The motor coils have a significant inductive component. Very loosely and from the point of view of a circuit only (not how the physics really works), inductors can be thought of as giving inertia to current.

Imagine what happens when a current has built up in a inductor, then a switch in series with it is suddenly opened. The current can't stop instantly (hence the inertia analogy), so will initially cause whatever voltage is necessary to keep it going. In this case, that will be causing a arc between the switch contacts. The greater this reverse voltage, the faster the current dies down. Eventually it gets to zero, the arc stops, and the switch if truly open. This may take µs to ms depending on the value of the inductance, the amount of current thru it, how fast the switch contacts can separate, etc.

The same principle applies whether the switch is a mechanical contact or a transistor. Transistors can turn off very quickly, so the high voltage caused by the inertia of the current the transistor is trying to stop builds up very quickly as well. It can easily build up to levels that will damage the transistor, or something else connected to the same line. The diodes provide safe conduction paths when this happens. They give the current a place to flow so that it doesn't have to cause a high voltage to keep flowing.

Another use for the diodes is to deal with permanent magnet motors that can be driven externally. Such motors work backwards as generators. If driven too fast, they can cause too high a voltage just from the generator action. The diodes again provide a path for this current to go so that the voltage can't get much past the power voltage. Note that this makes the motor essentially drive the power rail instead of the other way around. If the motor can be driven like that, this needs to be taken into account in the design of the larger system.