Despite the name, the Heli Pilot's helicopter does not have visible jet engines before or after purchasing this upgrade. This applies to every game that the Heli Pilot has been in.
However, most helicopters drive their rotor using what is known as a turboshaft jet engine. This is a jet engine similar to those on winged aircraft, but modified so that the engine's primary output is spinning a shaft rather than expelling high velocity exhaust. This shaft is connected to the rotor of the helicopter. Thus the Bigger Jets upgrade likely increases the size of this turboshaft engine, giving the Heli Pilot a much more powerful rotor.