Cyclaire's patented technology is, like many good ideas, based on a pretty simple principle.  In the case of Cyclaire its the combination of a double ended piston mechanism with a pull cord.

As you pull the handle, the pull-cord unwinds and, through a series of gears, drives the double ended piston mechanism.  The length of the pull-cord, the gearing and the double ended piston work together to mean that a lot of air can be pumped quickly.

The diagram shows what happens inside the Cyclaire pump. You can see how the piston changes direction as the system rotates.  In reality the piston moves much faster than is shown here, pushing large amounts of air into your tyre very quickly.