The phenomenon which you describe is known as a tidal bore. To understand it, one has to go beyond the strictly idealized linear theory for waves as taught in most courses of physics and consider waves that reach to a finite depth under the water surface. This is because the tug of the Moon on the water affects the entire ocean, from the surface to its very depth, and creates waves that may be seen as analogous to tsunami in the sense that they affect the entire water column (even though tides create much milder waves than tsunami).
The first step to the formation of a tidal bore is the approach of the very smooth, long-wavelength tidal wave to a broad shallow bay. As the wave enters the shallower water, it tries to conserve some of its momentum and energy by becoming steeper and higher, which also distorts the form of the wave. This can be derived from Stokes' wave theory and is known as wave shoaling. The shoaled wave is then further funneled into the river and can further increase the amplitude of the bore.
In an ideal set of circumstances, what forms in the river is called a solitary wave. Even though it is a solitary wave, it does not disperse due to interaction with the river bed, and this is the basis of the tidal bore. Some pretty fine-tuned circumstances are needed for a tidal bore, though, it will easily disperse at the bay, or the tidal wave simply isn't strong and fast enough to form a solitary wave in the first place. So you are lucky to have experienced it in your region!