An interesting discussion - which seems to be generating some passionate responses! And a topic connected to some research I had been doing for an article. According to the FAI, the body which is the keeper of such records, the Wrights are credited with "The first sustained, controlled, powered heavier-than-air manned flight". The question that has to be asked was what was different about their achievement which gives it the distinction.
Their flight was the culmination of a long development of flying machines which properly stated with George Cayley in the UK who was the first to appreciate birds flew due to the shape of their wings and not because of flapping. He also identified stability as a factor for flight and devised a control system. it was proved when his coachman (Cayley was not prepared to try it himself) was the first man to fly a controllable heavier than air machine in 1853. The problem with flight was power and 19th century experiments with steam engines produced powered short hops with one 3hp powered aircraft achieving an altitude of 6 inches.
The Wright brother's real achievement was developing an engine with sufficient power and to realise that sustained flight would require a control system. They had done a lot of research in their own wind tunnel and come up with a system which emphasised control but lacked stability. The engine produced 16hp at peak, but this was barely enough to get the flyer airborne. When they tried to fly subsequent versions without the strong winds of Kittyhawk, they needed a catapult assist, despite the engine having five more horse power. Only when they obtained 30hp could they achieve take off from level ground without assistance. By then, other people had achieved successful powered flight with different designs - largely due to the Wright's vigorous enforcement of their wing-warping patent. Ailerons (which had been first patented in England by Matthew Piers Watt Boulton in 1886) had become the standard, and with better stability characteristics.
There have been other claims to the title including Gustave Whitehead's, but it is the Wright's which is officially recognised. And that is important - just as Chuck Yeager's supersonic flight is recognised as being the first despite quite clearly not being so. (Big can of worms opened here!)