Instrument landing: why do all electronic devices need to be shut off?
Short answer: ILS is rather sensitive to interference and not all electronic devices take much precaution in avoiding the generation of interference. The pilot wants to be sure that the readings he's getting on the localizer and glideslope are accurate, since he can't actually see the runway to verify the final approach path visually.

Longer answer: ILS is actually relatively simplistic as RF signals go. Both the localizer and the glideslope just consist of a pair of AM signals that are broadcast directionally. For both systems, the modulated frequencies are 90 Hz and 150 Hz. For the localizer, 90 Hz is broadcast to the left of the runway centerline and 150 Hz is broadcast to the right of the centerline. When the aircraft is aligned with the runway, it will receive equal strength signals of both frequencies. When it's to the right, it will receive more power from the 150 Hz signal and when it's to the left, it will receive more power from the 90 Hz signal. The glideslope works pretty much the same way, just with the 150 Hz signal transmitted below the glide path and the 90 Hz signal transmitted above it. [1] [2]

Now, suppose some passenger's electronic device is emitting a frequency that it shouldn't be. This is not exactly unheard of among consumer electronic devices. This is more common among devices manufactured in countries with less-strict emissions testing requirements/enforcement, but it's possible due to slight defects in almost any device. [Source: I design RF receivers for a living.] Now suppose that this noise happens to be on the localizer or glideslope frequency for the runway to which the aircraft is flying an approach. Hopefully you're starting to see a problem here... Whatever amplitude changes are happening on this unintended emission are now being received by the AM demodulator of the localizer or glideslope, potentially confusing it and, in low visibility conditions, causing the aircraft to be directed somewhere other than the runway threshold. Needless to say, that's bad.

Boeing recommends the following actions for flight crews flying ILS approaches in order to detect erroneous ILS readings:


Crosschecking altitude and DME distance periodically.
Crosschecking altitude and flight management system (FMS) threshold distance.
Crosschecking altitude and the crossing altitude of the outer marker (or locator, very-high-frequency omnirange [VOR] navigation equipment, or FMS).
Crosschecking radio altitude and barometric altitude.
Crosschecking ground speed and rate of descent.
Questioning air traffic controllers when indications do not appear to be correct.


However, even when following those instructions, slightly erroneous ILS signals may be hard to detect until too late. If a DME is showing that you're half a mile off, you can probably notice that, but a couple hundred feet off might not be noticeable (but is still plenty to be a big problem.)