Sports cars use the same basic processes as other cars: a powertrain supplies drive, tires transmit forces to the road, suspension controls wheel and body movement, and brakes reduce speed. What makes the result feel sporting is how those systems are selected and coordinated around acceleration, cornering response, and driver involvement. There is no single engine size or layout that every sports car must use.
A light roadster and a powerful grand tourer can achieve that goal in different ways. To understand either one, follow the forces through the whole car rather than starting and ending with horsepower. Vehicle mass, the location of major components, gearing, tire characteristics, and chassis settings all influence how a driver's input becomes motion.