Is it just a coincidence that the moon's period of rotation and revolution are identical, so that we always see the same face? If not, how did this situation come to be?

"The moon keeps the same face pointing towards the Earth because its rate of spin is tidally locked so that it is synchronized with its rate of revolution (the time needed to complete one orbit). In other words, the moon rotates exactly once every time it circles the Earth.

"The same forces that create tides in the Earth's oceans (from the gravitational pull of the moon and, to a lesser extent, the sun) also act on the solid body of the moon. The Earth's gravitational force on the moon distorts the moon into a slightly prolate, or football, shape; in addition the moon's intrinsic form is somewhat egg-shaped. If the tip of the football/egg does not point toward the Earth, then gravitational forces exert a torque that makes the tip point back toward the Earth (in reality, the moon oscillates a small amount around perfect alignment, a motion called the lunar libration)."

It is very unlikely that the moon started out synchronized; that would indeed be a surprising "coincidence." As Boss explains, "The moon's synchronous spin state is thought to have arisen billions of years ago, when the moon was much closer to the Earth, and so tidal forces were much stronger than at present. The Earth's gravity maintained this spin state even as other gravitational interactions caused the moon to move outward to its present orbital radius.