FM works by varying the frequency of the signal around the nominal carrier frequency. Because the frequency is varying, the signal is not a pure sine wave. Therefore, it necessarily has some energy in sidebands as well as the instantaneous frequency. The higher the audio frequency, the more the signal deviates from a sine wave, so the more energy ends up in the sidebands instead of at the instantaneous frequency.
If one works out the math, one finds that a FM signal has an infinite bandwidth (sidebands of arbitrarily high frequencies), but the energy drops off quickly as we look farther away, so in practice they are negligible (and may be filtered out).
3 kHz is the deviation of the signal — the amount the instantaneous frequency differs from the carrier frequency. Because there are sidebands, the occupied bandwidth is larger than the deviation.
If you have access to a software-defined radio or panadapter, try tuning it to the wideband FM broadcast frequencies. You'll see that even though the bandwidth of the signal is big enough to easily get a high-resolution look, it doesn't look like a single instantaneous frequency but a wide hump. These are the rapidly varying FM sidebands. In particular, when the station is quieter you may be able to distinguish a set of three or more peaks moving in concert — these are the result of the inaudible 19 kHz stereo-encoding “pilot” tone which is always present for a stereo transmitter.
“Why are audio levels measured in kHz?” Because FM maps the audio signal to frequency changes, so the only absolute measurement of the level on the air is the change in frequency. Any other way of measuring audio — sound pressure level, AF voltage — is inapplicable because those depend on the characteristics of the transmitter or receiver.