Any wire acts as an antenna - if you're really unlucky it can even amplify the signal.
You've probably heard the kinda "morse code" sounding interference you get on headphones when a mobile phone is ringing. This is an example of the headphone cable acting as an antenna.
When aircraft systems communicate with each other (e.g. the antenna for an ILS sending a signal to the display on the flight deck) they send the signal down a wire. If a mobile phone is transmitting (which they do all the time when switched on) this signal can interfere with the desired ILS signal. The chances of it happening are remote, but that interference could, in the ILS example, fully deflect the CDI, making it useless.
As aeroplanes become more and more electronic, the opportunities for such electromagnetic interference to introduce gremlins increase. Basically any electronic system could go wrong. It's just a question of probability. The further away the source of interference is, the less the chance of it having an effect. However if you placed a phone (or other transmitter, such as wi-fi device) next to a navigation radio, it could tell you that you are a long way off course.