What is the "sports" in "sports car"?

Yes, motor racing is a sport and that's where the "sport" comes from - sports cars are designed primarily for speed/performance instead of comfort/space/practicality/economy.

Sport originally meant fun, pleasure, amusement, play. This was the sense of the original "Oh what sport it is to ride and sing" in Jingle Bells (which most people know as the later amendment Oh what fun).

So no – a sports car isn't particularly a car designed for or derived from the "sport" of motor racing. It's a car designed to be "fun to drive". Which in practice, as @Waggers says, means designed primarily for speed/performance instead of comfort/space/practicality/economy.

Although I would say that actual speed/performance is not always a paramount consideration for sports car designers. Their target customers aren't really likely to driving competitively; many of them will be far more concerned with the appearance of high performance than the reality.

Waggers has it. "Sports" cars are designed for speed and performance, and originally got the name because they evoked images of, or borrowed design and engineering concepts from, dedicated racing cars. "Factory racing teams" have been a mainstay of the auto industry since the Great Depression, and as the sport became popular, people wanted machines that had similar performance.

Because speed and performance generally require very expensive engineering and components (large-block engines, multi-stage turbos or superchargers, double-wishbone suspensions with 4-wheel struts, ventilated brakes, low-aspect tires, alloy rims, etc etc), to make the car look like it's worth as much as it costs to incorporate these features, most makers also throw in much less expensive-to-include luxury touches, such as leather, premium sound systems, wood trim, "power everything" (windows, locks, seats, mirrors), and now GPS navigation, memory seat positions, keyless start, etc etc. BMW, Mercedes and Jaguar were early proponents of this "luxury sport" concept, and their rise to popularity in the U.S. eventually convinced the Big 3 to change their luxury car lines as well.

Sport is a noun and a verb; as sports it is used also as modifier in phrases such as "a sports center."

Sport has also the dated meaning of "entertainment, fun."

It was considered great sport to trip him up.

I wonder if the term orginated with an old meaning of the term sport, describing a flashy person attracted to pleasurable pastimes. It may not have been sports car so much as a sport’s car — indicating that a person who owned such a vehicle would be considered a sport.