Why can a Pilatus PC-24 land on unpaved and short runways like no other business jet?
The Wikipedia page on the Pilatus PC-24 gives you some hints:


  The PC-24 is a twin-engine business jet, a larger jet-powered follow-on to the earlier Pilatus PC-12. It is considered to be a short takeoff and landing (STOL) aircraft, being designed to operate from short and rough airstrips. It incorporates an advanced wing design, with a large double-slotted flap system to achieve the necessary performance, having a stall speed of only 81 knots at the maximum landing weight. Pilatus has claimed that the PC-24 possesses performance attributes which are unmatched by any jet aircraft on the market. It possesses a takeoff distance of 820 m (2,690 ft) and a landing distance of 770 m (2,526 ft).


...


  The PC-24 is powered by a pair of Williams FJ44-4A turbofan engines. [...] The engines are mounted on the upper portion of the rear fuselage to prevent foreign object damage (FOD) during operations at rough airstrips. Other FOD-reduction measures include the enlarged flaps and chines on the nose wheel of the landing gear. The exhaust ducts of the engines feature an aerodynamic modification, having been shaped so the outflow contributes to the thrust vector during takeoff, reducing the takeoff roll distance.


So, in summary:


Advanced wing design with double-slotted flaps
Low stall speed
High mounted engines
Large flap and main nose wheel chines
Reduced take-off roll through vectored thrust
(Thanks to Peter Kämpf) Lower Wing Sweep to add effectiveness at low speeds at the expense of cruise speed.


There really isn't a lot that is going different from other business jets other than a shorter take-off/landing roll. The design isn't dramatically different, and it doesn't mention it but it probably has reinforced landing gear. 

No production PC-24's have been delivered as of yet, so its hard to tell what they look like. The test PC-24's only have 150 hours behind them so they may not have had any rough-strip experience. It's difficult to say exactly what the modifications are because they may include other things like FOD deflectors on the wheels.