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The inflatable seat has a pocket for the Wii remote, which is used to sense movement via the accelerometers. Players hold the nunchuck to simulate holding the reins or, as the Nintendo patent describes, “a raised hand for balance as in Bronco riding, a lasso, a sword or other weapon, etc.”
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The patent, found in the European patent database, gives an example of a possible game.

For example, the Wii main unit can display the back of a horse’s head and background of a trail, a field, a battlefield or other environment the virtual horse is being ridden through (...). The image may change in response to detected motion of the user.

But, wait the seat isn’t limited to horesback riding! Nintendo says their inflatable seat can be used to simulate riding other animals. The patent lists “bulls, camels, elephants, burros, dolphins, whales” and imaginary creatures such as “dragons, griffons, unicorns, giant eagles” as possible mounts. Nintendo also suggests the seat can be used for vehicles too like “hang gliders or other aircraft, a magic carpet, a motorcycle, a jet-ski, a land or sea exploration vehicles.”