The first airplane by the Wright brothers had a double wing. This concept continued through World War 1 and into the 1930s - why was this concept popular?
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.
In the early days, it really looks as if people hadn't really worked out what worked best
A wire-braced box girder structure can probably provide better strength and rigidity than a single, and much longer, wooden spar.
An advantage of multiple wings is that you can have shorter wings, and this can improve maneuverability in combat
Maybe a better question is why people opted for only two wings when they could have had twenty?
The main reason for having multiple wings in the initial years of the aviation was the lack of availability of materials with sufficient strength.
The main advantage of the biplane is that the wings could be shorter for a given lift. During the initial stages of aviation, the materials available for manufacture of aircrafts were timber, doped fabric etc. The structural members made of these did not have enough strength, limiting the wing size.
The early aircraft also had less powerful engines, resulting in lower speed. In this case, the lower stall speed of the biplanes was advantageous.
The biplanes had very good maneuverability (better roll rate) compared to the monoplanes. This is the reason triplanes were developed during the WWWI.
"SopTri3" by http://www.earlyaviator.com/archive/a/images/tripe_peggy.jpg. Licensed under Public Domain via Commons. However, with the advent of better materials like Aluminum alloys, the aerodynamic disadvantages of biplanes became apparent and monoplanes rapidly became the preferred design choice.
As the material strength improved, the aspect ratios have become ever bigger, with civil aircrafts facing operational (airport) limitations rather than strength limitations in wing length.
However, the biplanes are still used. The better maneuverability and low stall speed make them popular aerobatic aircraft.
Actually, there were quite a few monoplanes around even in the early days:
All images from Wikipedia
Even though some early aircraft were designed with cantilever wings which did not need wire supports, there were still a number of reasons why the wire truss biplane design remained in use for a long time. Like any new technology, development takes time. Fokker designer Reinhold Platz' designs during WW1 is a great example. His thick wing, box spars were a sound design, yet glue was not necessarily very strong, and/or the skilled labor for assembly was poor which created problems, and deaths. The wood is not the problem. High quality spruce actually has about the same strength to weight ratio as aircraft aluminum, though the AL is more isometric, wood proved itself worthy of aircraft construction when used correctly. When the Fokker box spars failed, word got around fast, and generated fear and loathing by Pilots who liked to know their wings would not fold. Pilots didn't dislike biplanes, and wanted "what we know already works". The triplane had box spars and were true cantilever wings without braces and struts, yet pilots veered away from it in great fear, and refused to fly it. So, Platz added the interplane struts to tie all three wings together at the ends purely for the impression it made on pilots. Wires were however eliminated. The Fokker D8 monoplane had a true cantilevered wing which also had some quality issues early on which of course usually ends in a death, and pilots didn't like that. Wired monoplanes had no advantages, High drag, lower lift and weaker than biplanes. Pilots, passengers and investors liked biplanes. Designers eventually proved their knowledge to pilots and to the public, but like anything else there is an enormous psychological element- A not altogether invalid fear that slowed progress. Military people are the worst at accepting new ideas, with a long history of accepting only the tried and true, with no real desire to put pilots at unnecessary risk. Plus there were two distinct advantages to biplanes, adequate lift with shorter wing span, plus the associated high roll rate, and strength was, and is- very good, they were in fact hard to beat.
Structural strength was not the main reason for early bi-planes. Otto Lilenthal did his glider experiments in a monoplane like structure. The wright brothers were following his research but they wanted to create powered flight. The earliest engine didn't have sufficient power. A Biplane could generate more lift at slower speed. A monoplane couldn't. That was the primary reason for choosing a biplane design.