Question: Why do most performance cars have a longitudinal-mounted engine?

By performance cars I mean sports cars that typically develop at least 400hp and are usually coupes two seaters.

Answer: 
It's quite simply because a longitudinally mounted engine equates to less of an engineering challenge to achieve "balance". Imagine something like a large V formation engine, two banks of cylinders, two sets of exhaust headers, etc... The simplest way to divide this large heavy lump of metal and gaskets is along the centre-line of the crank. If that is aligned with the centre of the vehicle it is installed in, the vehicles balance left to right is near perfect.

Add a gearbox to the end of a longitudinally mounted engine plus a diff and prop-shafts that are all inline with the centre of the vehicle and you still have not only near perfect symmetry but likely equal length driveshafts. This is important as it avoids torque steer. With a 100bhp engine, torque steer is negligible, quadruple that figure and it becomes a big issue.

In transverse (typically FWD applications) it is usual to offset the bulk of the mass of the engine over to one side of the chassis and place the gearbox on the other side of this. At normal road speeds and relatively low power figures, this is a good compromise as it allows maximum cabin space. At higher speeds however and higher power figures, it significantly compromises the handling and indeed safety of the vehicle.
And to add to @MeltingDog, there is also the concept of weight distribution - some cars even have the engine at the front and the gearbox / rear axle at the rear to improve the distribution.