How does this "immobilizer" immobilize the car? By shutting off the fuel pump?

you are correct. the immobilize cuts off the fuel pump. the RF signal from the key is transmitted to the ignition cylinder where the transponder RF ring sits. From there, data is transmitted to your data system. from the data, it is sent back to your fuel pump. There is no actual way to bypass an RF data system. Unlike GM where its all resistor based, hacking it is easy. as for the cellphone issue, the frequency of the cell should not affect the cars immobilizer.

do you have an aftermarket alarm on the car?
does the car ignition come on. im asking because fuel pump gets it power from the ignition 2 wire.