First of all: aviation was being researched around the globe, and there is still some debate about who was first to achieve powered flight. Nearly every western country had their own aviation pioneers. Louis Bleriot in France to name one.

Military aviation started with the use of balloons for observation/reconnaissance, firstly in the Battle of Fleurus in 1794, so the use of aircraft for military purposes was not a new idea. Airplanes were first used in the exact same role as the balloons, but during wwI were developed into weapons. First pilots/observers simply brought weapons with them, and dropped grenades over the side of the aircraft, later on weapons were fixed to the aircraft culminating in air to air combat and bombing.

The Wright Brothers were definitely aware of what they did, between 1906 and 1907 they tried to convince several governments they built a working flying machine. At first the US government payed no attention, and they took their invention to France. Upon their return, after they spoke to Lt. Frank Lahm, the us signal corps issued specification no 486 calling for a military aircraft, but there was no existing aircraft that fit the specification.

Then they basically got bogged down in a patent war, as several manufactures, most notably Curtiss, circumvented the wright patented design for steering the aircraft along the longitudinal axis by warping the wingtips. Curtiss designed the aileron, which is not a warping wingtip, but a control surface which accomplishes the same. By 1912 the wright designs were outdated, mostly because they were focussed on fighting their patent wars and didn't develop their designs. In short: they failed to capitalise on their invention, not helped by the fact that there were several contemporaries trying to do the same thing. They did win the patent war with Curtis eventually though, in 1913. Too little too late.

I'm an undergrad history student/former airline pilot.