Why do older airplanes have a wing on both the top and the bottom of the aircraft?
Mainly for strength. Up to about 1920, wings used very thin airfoils and were mainly made of fabric stretched over a wooden frame. The main spar was too thin and would have been prone to bending if it had had to support the whole weight of the aircraft in a single span. Only by using two wings, the upper as the compression member and the other as the tension member of a truss, the needed strength was possible.



Airfoil thickness comparison by D.R. Kirk, Florida Institute of Technology (picture source)

In your photo, you can just about see the wires running from the top wing (near the outboard struts) to where the bottom wing joins the fuselage. These wires take most of the strain in flight, spreading the load on the wings. It's the same principle as using triangles in the construction of electricity pylons or bridges.

Monoplanes of that time needed even more wire bracing. Look at the picture of a replica of the Etrich Taube, a very popular plane of the pre-WW I period.



Etrich Taube in flight. Note the truss below the wing and the many wires keeping it in shape (picture source).

The shorter wingspan of a biplane also reduces the load on the wings. Biplanes tend to have a lot of lift (for their size) because of the large wing area, but also a lot of drag, so they're quite inefficient overall. There were also triplanes, with three mainplanes for even more lift in the same wingspan. WW I pilots demanded the highest rate of roll, which could best be achieved with biplanes. 

In the time between the World Wars, the use of high-strength aluminium for aircraft and the monocoque airframe both allowed airframes to become stronger. This meant that monoplane designs became more practical, even though the older aviators didn't believe a monoplane could be strong enough. As soon as monoplanes were practical, the improvements in efficiency and flight performance saw them replace bi- and tri-planes.