The hyperdrive engine operates by highly charged plasma from a ship's fusion reactor being channeled into the hyperdrive ring around the spacecraft to create a high density rotating magnetic field that propels the craft into a different dimension, allowing them to take a "shortcut" through normal space without actually exceeding the speed of light.
Speed while in hyperspace is measured by a factor of the original hyperdrive "speed". Hyper +1 is roughly equivalent to traveling at the speed of light (as noted above, the ships actual velocity does not exceed light speed, but by taking a shortcut through normal space in the hyperspace dimension, a ship traveling at Hyper +1 would arrive at it's destination about the same time a beam of light would.) The scale then expands exponentially.
(Refer to the below diagram)
Deuterium fuel is fed into the fusion reactor.
The reaction produces heat and highly charged plasma.
Heat rises to the top of the reactor and goes through a heat exchanger. This heat is used to heat the ship's interior, and to heat water for the crew for showers, cooking, etc.
The plasma goes out through the plasma manifold and through a magnetic turbine. This turbine outputs electrical power for the ship.
The plasma then goes out to the plasma coils outside the ship.
The plasma coils produce a high-density magnetic field.
The high-density magnetic field can be manipulated to move the ship at sub-light speeds, or, at high-output, to move the ship into hyperspace.
Lowering the plasma flow reduces the strength of the magnetic field, dropping the ship back into normal space.
(click for larger version)
Behind the scenes information:
The hyperdrive system used in the novel is based on the theoretical work done by Burkhard Heim in the
1950's that suggests that a rotating magnetic field might overcome gravity enough to propel a spacecraft.
It is doubtful however, that the speeds attained would be anything close to what's used in the novel.
Also see the following articles for more information on Bukhard Heim and his theories: