I agree that you must be talking about music file resolution, because 384kHz is one of those magic numbers used for resolution limits. However, your question as paraphrased is worth a response - why is it important that an amplifier have a bandwidth greater than 20kHz?
It's because the nature of electronic amplification produces a bandwidth of response. In our subculture, that all-important bandwidth is 20 to 20 kHz. However, when looking at bandwidth of an amplifier, it doesn't fall off a cliff at the 20 and 20kHz ends. Instead, it usually produces a logarithmic curve. It's why amplifier bandwidths are specified in +or- dB variance from that line of 20 to 20kHz. Typically, if an amplifier bandwidth is 1 or 2 dB down at 20kHz, it's a half dB down at maybe 15kHz. That has a potentially audible effect. So, it's advantageous to push the ends of an amplifier response well beyond that audio band. That helps to ensure any effects of that response decaying down is well beyond what you would be able to hear.