From top to bottom:
1. The piston heads are getting farther apart and drawing air in through the hole
2. The green piston head passes the right hole and no more air can move in or out of the space between the piston heads.
3. The piston heads have squeezed close together and the red head has past the left hole so air can now escape.
4. What little air is left between the heads is reduced in pressure as the heads move apart.
5. Air rushes in to fill the partial vacuum between the heads through the hole.
The effect is to pull air in through the right hole and push it out through the left one, but the advantage of this design over others is that valves do not have to be opened and closed mechanically along with the motion of the pump. This should enable it to pump faster and generating less heat.... Maybe valveless isn't the best description as the body of the pistons might be considered as acting as a sort of valve, but oh well.