Why do aircraft with turboprop engine have black painted anti-icing system?
These are rubber boot deicing systems. When ice forms on the leading edge of a wing, pressurized air is used to inflate the boot so the ice will pop off. Normally, this inflation is not permanent but the air is pulsed. Activation is done by the pilot, so this system is normally switched off. Operation needs some care because if used too late (with too much ice) the boot may become impossible to be inflated.
Historically, it was believed that activation with too little ice would also render the boot ineffective, with ice building up around the inflated boot (ice bridging). This seems to have been a myth.

Deicer boots in operation (source: Wikipedia)
Other deicing systems use heat or constantly seeping deicing fluid. The first method is normally used on jets with bleed air or electricity as the heat source. The second system was for example used on the Beech Starship which had a titanium leading edge with millions of tiny, laser-cut holes for the application of deicing fluid.
Alternatively, surfaces are designed large enough to still be effective under icing conditions.
Fast jets do not need a deicing system - their stagnation point temperature is sufficient to remove ice. However, ice can still be a problem: When the supersonic B-1A was converted into the more stealthy, but subsonic B-1B, ice buildup inside the intake (which now had obstructions added so the engine compressor would be hidden from radar) would result in large chunks of ice building up, then breaking loose and damaging the compressor blades.