How exactly does nonlinear quantization in compading help in data compression
The human ear has a surprisingly large dynamic range.  We can hear very quiet sounds and very loud sounds.  If you have a microphone generating voltages based on the volume of the sound, and then you quantize the voltages, you would need a rather large dynamic range.  According to what you read, it would take 13 bits to make sure that you can properly capture the human ear's ability to detect loud and soft sounds.

However, we can observe that the human ear's sensitivity to volume is not linear.  As we see higher sound pressure levels (SPL) corresponding to louder signals, we find that the human ear is less sensitive.  Thus, if we are hearing a sound whose SPL is 0.006Pa (which is a 50dB sound), a difference in pressure of 0.001Pa is extremely noticeable.  If we're hearing a sound whose SPL is 6.3Pa (110dB), a difference in pressure of 0.001Pa is hardly noticeable at all.

Thus, if we're encoding an audio sample for human consumption, it's wasteful to simply quantize the sound pressure level (which typically corresponds to voltage in the microphone) linearly.  However, if we encode the sample non-linearly, we can provide finer resolution of pressure levels when the sound is quiet, and a coarser resolution of pressure levels when the sound is loud.  Because of the limits of the human ear, the consumers of this audio sample won't notice that loss of resolution as the sound gets loud.  As it turns out, 8 bits can encode sound pretty well for the average consumer of audio, though higher bitrates are common for those who seek higher fidelity sound.

We can't always use these techniques to encode the sound.  Recording studios will often record at a much higher bitrate, such as 16-bit or 24-bit sound and then mix it down to 8-bit later.  They do this because they will process the sound through many filters, and some of those filters can bring out the distortion that arises from a 8-bit nonlinear encoding.  Once they're done running it through the filters, they can downsample it to 8-bit using a common nonlinear function if they wish.