The mechanism is a latch. A peg on the prosternum sits against the lip of a cavity on the mesosternum. The beetle arches its body, loading the muscles against the latch, then the peg slips and drives into the cavity, producing the click and an abrupt rotation of the body. Acceleration during the launch is extremely high, reported in hundreds of times gravity for small species, because the energy is released far faster than muscle alone could deliver.
The jump is not accurate. The beetle spins in the air and lands more or less at random, so several attempts are often needed before it lands upright. That is enough, since the function is righting and escape rather than travel.
The eyespots serve a separate purpose. They are pigment markings on the pronotum, not eyes, and are generally read as a deterrent that suggests a larger animal to a predator viewing the beetle from above.