The real reason is that the signals generated by your radio receiver (yes, it generates signals as well as receives them) can interfere with the aeroplane’s navigation equipment.
In an article for ‘The Straight Dope’, published in 1987, Cecil Adams (who ran a similar, but far superior, column to this one) explained it far better than I could. He said,
“Most modern receivers use something called a “local oscillator,” which is sort of an internal transmitter. The oscillator generates signal A, which is mixed with the somewhat raw incoming signal B to produce nice, easy-to-work-with signal C. There’s usually some sort of shielding around the oscillator, but it’s not always effective and sometimes errant signals leak out to make life difficult for other radio equipment nearby. If the other equipment happens to be an aircraft navigation device, somebody could wind up digging furrows with a $25 million plow. So do your bit for air safety and bring a tape player instead.”