I believe the natural oscillatory period of the water in the Bay of Fundy matches that of the local tides, so that a state of resonance exists, like pushing a child on a swing in time with the period of the swing.

Resonance, just like diving from the rocks to the narrow channels from the bay in Alcopulco, where the waves fill the cut as you fall, or the wave runs up the river from San Francisco Bay for miles.

Yes, I've always heard this as an example of tidal resonance or tidal locking, though I've never been there, the idea certainly makes sense. A wave moving across the bay has a natural frequency, i.e. of a wave leaving the shore and returning. This matches the natural frequency of high tides, i.e. about once per half a day, so we get resonance. To put it another way, there's a high tide, so the water is high at one side of the bay. The Moon moves on (the Earth spins), the wave leaves. By the time it returns, there's another high tide (the Moon is on the opposite side of the Earth as the Earth span), so little energy is lost in creating the next high tide. The energy from this new high tide is then added to the old, resulting in a large amplitude. Cool stuff.

The bay allows a 'seiche' to form, a form of resonance. You can create this effect in your bath tub by sliding back and forth in synchrony with the natural frequency of the water. Within a very short time you will have increased the amplitude enough to have a very satisfying wave of water pouring over each end of the tub. As described by other answers, the tidal rhythm and the moon's pull work to create the same effect in the Bay of Fundy by having the same frequency.

To understand what happens in the Bay of Fundy, start with a hand basin half-full of water. Push down on the surface on one side with the palm of your hand and the water will rise on the other, after which it will slosh back and forth like a liquid see-saw. By pushing down even more on each side at the same time as the level on that side is falling, the rise and fall of the surface will increase - and can be made to overflow the rim of the basin. The sloshing of the water has a natural frequency and your additional input resonates with it, increasing the amplitude of the see-saw wave. At the central axis of the basin the level remains unchanged, although water moves to and fro horizontally.

Now imagine the basin cut vertically down that central axis and consider just half of it. The half that is left corresponds to the Bay of Fundy, the axis-edge marks the opening of the bay, and the missing half of the basin is replaced by the open Atlantic Ocean. An incoming tide effectively appears in the ocean as a huge wave advancing towards the bay. As it reaches the continental shelf at the bay's opening it plays the role of the high half of the see-saw and happens to coincide with the low water level at the far end of the bay. By the time this wave has moved to the innermost section of the bay - raising its water level to a peak - the dip in the ocean surface corresponding to the low tide has reached the continental shelf.

The exceptionally high tides occur because the successive incoming tides appear at nearly the same frequency as water sloshing into and out of the bay, just as happened in the wash basin. It is a resonance effect.

If the Bay of Fundy opened directly into the Atlantic it would not have such high tides because the natural period of water moving to and fro in it would be only about 9 hours, which is not close enough to the 12.5-hour period of the tides to lead to significant amplification of the wave motion. However, the Bay opens into the Gulf of Maine and together they have a natural frequency of just over 13 hours.

Richard Holroyd, Cambridge, UK