They're used mostly on motorcycles shows now, but 45 years ago, when an average racing bike wasn't that impressive and usually powered only by a 100-250cc engine, Hornet, with its 654cc engine, lightened bodywork (still could be lighter though) and the exhaust system without mufflers (this explains the name "Spitfire Hornet", used till 1966) giving a few more hp, could provide required power and was able to withstand the toughest trails. And that's exactly what the American racers needed at the time.
Designed exclusively for an off-road use, BSA Hornet first came on the US market in 1964. They didn't have a head-light or a tail-light, as well as the turn indicators. Produced to be a desert racer, the bike simply didn't need all this. Although BSA had foreseen a possibility to convert Hornet into a bike that could have been ridden on a highway, by hooking up the proper illumination and the mufflers. No wonder, that tons of them have been successfully equipped with the lights and adapted for the road use. That's also how you can often see them on the bike shows nowadays.
BSA exported Hornets only in the US. This particular bike in the picture above most likely came out in 1967 and, taking into account its color and high mounted exhaust pipes, was designed for the east coast. The thing is that "Cherokee Red" color was chosen over "Mandarin Red" only in 1967. Also, BSA supplied bikes with low TT pipes to the west coast.
Needless to say that Hornets were very desirable among dirt bike riders. One of the most popular celebrities at the time and "racing adventures" fan, who rode a Hornet was Steve McQueen, well known by his roles in The Sand Pebbles, The Thomas Crown Affair, Bullitt, The Getaway, and Papillon.
Hornet had its time and recognition, but the production was closed in 1967.