You have to turn off electronic devices during take-off and landing (cell phones, wireless computer mouses, etc are banned for the duration of the flight) because they pose a risk called electromagnetic interference.
Electromagnetic interference is experienced by all of us on a regular basis. An example of this is if you put a cell phone near the computer, you can hear loud static in the computers speakers every time the phone rings, and the screen may start to shake. This technically should never happen, but the wire to each speaker is acting like an antenna, and it picks up side bands in the audible range. This is not a dire problem -- just a nuisance. But notice how common it is. In an airplane, the same phenomena can cause big trouble.
An airplane contains a number of radios for a variety of tasks. There is a radio that the pilots use to talk to ground control and air traffic control (ATC), a radio that the plane uses to disclose its position to ATC computers, there are radar units used for guidance and weather detection, and so on. All of these radios are transmitting and receiving information at specific frequencies. If someone were to turn on a cell phone, the cell phone would transmit with a great deal of power (up to 3 watts for a single phone). If it happens to create interference that overlaps with radio frequencies the plane is using, then messages between people or computers may be garbled. If one of the wires in the plane has damaged shielding, there is some possibility of the wire picking up the phone's signals just like a computers speakers do. That could create faulty messages between pieces of equipment within the plane. Now imagine what would happen if everyone on the plane were to use electronic devices, and you should be able to fully understand the ban on such devices.
Of course having a bunch of people on their cell phones at the same time would be annoying, and would make it harder for people to hear instructions should an emergency arise, thus it is another reason for the ban!
****EDIT****
The reason you can have it in "plane mode" during the flight (only if your phone has that capability) is so that you can listen to music or play games. This effectively shuts the phone down from trying to send and receive transmissions which could potentially interfere with the equipment on the plane. ******
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