Let me see if I can explain this without getting too technical:
Most analog radios use a technique called frequency modulation (FM), which works by putting up a signal called a carrier and changing its frequency in a way that tracks the wave shape of the sounds you make. This is the same scheme used for broadcast FM, but some of the numbers are different.
Have a look at the first attached picture, where you'll see a couple of cycles of a sine wave. If you were listening to this as sound, you'd hear it as a tone. You'll notice that it has three features: peaks, troughs and places where it crosses the zero line. When the wave is at zero, the transmitter puts out an unmodified carrier. (This might be at, say, 462.55 MHz.) As the wave's amplitude (which is like its volume) changes relative to the zero line, the transmitter changes the frequency of the carrier in a way that matches it. If its position is positive, the frequency increases; if it's negative, the frequency decreases. The amount of this movement is called deviation. When the wave reaches a peak or a trough, the deviation is at its maximum in one direction or the other.
Radio channels are carved out with a certain amount of maximum deviation in mind. For example, the GMRS channel at 462.55 MHz is a "wide" channel designed for radios with a maximum deviation of 5 kHz (0.005 MHz, to keep everything in the same terms). This means that a properly-functioning radio won't deviate any more than that when the audio is at its loudest. If you do a little arithmetic, you'll find that this channel actually occupies the range of 462.545 to 462.555 MHz, which is the center plus or minus the 0.005 MHz worth of deviation. There's also a small empty area between channels called a guard band that's used to keep radios using adjacent channels from interfering with each other. "Narrow" channels have a maximum deviation of 2.5 kHz (0.0025 MHz). The shared GMRS/FRS channel at 462.6125 MHz is one of these channels, and it occupies 462.61 to 462.615 MHz.
Here's where the fun starts:
A narrow receiver is designed with the expectation that the signals it receives will only deviate by 2.5 kHz, and that it should represent a peak or trough when the signal deviates that much. So what happens when the signal deviates by more than that? Well, the circuit that makes the audio representation can only go so far, and it hits the wall in one direction or the other, even as the deviation continues to increase. The result is called clipping, which is illustrated in the second picture. You'll notice that the wave doesn't have nice, round peaks and troughs like the sine in the first picture; the tops are chopped off. The result, when converted to sound, is nasty, distorted audio. As the deviation increases beyond the maximum, the shape of the wave gets even more chopped off, and the audio gets worse.
What this means for you, Dan, is that your radio is either programmed to operate narrow or the audio you're feeding it isn't strong enough to make it deviate past the point that would cause clipping in a narrow receiver.
Hope that explains it.
--Mark