Every time you pull Cyclaire's handle, the pull-cord unwinds inside and drives a pulley mechanism. The pulley is attached to a double-ended piston, so that the piston moves backwards and forwards inside the cylinder while the pulley is turning. This is how the Cyclaire Bicycle Pump pushes so much air into the tyres; the further you pull the handle, the more times the pulley turns and the more air the piston pushes into the tyres.
In general a bicycle pump functions via a hand-operated piston. During the up-stroke, the piston will draw air through a one-way valve into the pump from the outside. During the down-stroke, the piston then displaces the air from the pump into the tyre. Unlike normal hand-held pumps which have a single-ended piston, the double-ended piston inside the Cyclaire combined with the long stroke obtained through the pulley system provides the speed.

The image below shows what's happening.
The piston changes direction, feeding air into the hose. The piston moves very quickly creating a lot of air which is how Cyclaire pumps more air per stroke than any mini-pump, and inflates your tyres and inflatables to high pressures more easily and quickly.
The new Cyclaire Rapid has a different gearing ratio to the Cyclaire Plus, so it pumps 50% faster! It inflates to 80psi so it's ideal for mountain bikes.