According's to Joule's law of heating, the amount of heat developed per second by a wire of resistance R due to the flow of electric current is
H=I2R ⇒ H ∝ R, for constant I.
So, the heat developed across the wire depends on its resistance. As the resistance of the filament of the heater is very higher than that of connecting wire, the heat developed in the filament of the heater is very high and becomes very hot whereas the heat developed in the connecting wire is very low and is not heated.