Inductive sensors work based on two very important phenomenon viz. resonance and eddy currents. The following figure presents a block diagram of a commercial inductive proximity switch.
The coil and the oscillator are at the heart of the inductive sensor. The oscillator is commonly a RLC resonance circuit. As a metallic target is moved towards the detection coil the amplitude of the oscillations in the coil begins to decrease, since eddy currents are setup in the target, which load the coil. This is detected by the trigger circuit and it sends a signal to the output block that a target has been detected. A schematic of a very crude proximity switch is presented in the figure below.
As a target is brought near inductor L1 the amplitude of the oscillations decreases, which is detected by a Teensyduino that switches on a LED.