What is the physical significance of the Fourier coefficients in the DFT of audio signals and how can they be best displayed in a spectrogram?
This is a two year old question, but I think I add a little bit for those who might find it doing searches.

When you are talking about a spectrogram, there are really two steps involved that have to be considered.  First is converting from the signal to DFT bin values, and the second is displaying those bin values in a pixel display.

Ideally, a spectogram of a pure tone of increasing frequency should make a solid, non varying, upward sloping curve.  Inherently DFTs don't work this way.  When you are at a whole integer frequency (in terms of cycles per frame), there is a single bin value, the rest are zero.  When you are halfway between integer frequencies, the bin values are spread out which is commonly known as leakage.  My recommendation is to use a VonHann window on your signal before you do the DFT.  Depending on your scaling, this is the equivalent of subtracting the average of the two neighbor bin values from each bin value.  This has the effect of "tightening up" the leakage, and "spreading out" the sharp peaks.  This makes the display look a lot more consistent independent of the value of the frequency compared to the frame length, which is what you want.

The phase values in the DFT are framing dependent, and therefore don't help much with the display.  Instead you should concentrate on making an intensity display based on the magnitude of the bins.  You are not limited to a linear mapping, or a log mapping.  You are going to want to use a lookup table anyway to speed up processing, so you can pretty much define an monotonically increasing function that works the way you want.  Use can use a monochromatic color scale, or changing color one.  The standard green for low, yellow for mid level, and red for intense is easily comprehended.

The second choice you need to make is the vertical axis scale.  The two logical choices are linear, which the DFT is inherently. or logarithmic corresponding to musical scales.  In either case, you are going to need to map a bin, or a set of bins, to a pixel, or a set of pixels, depending on the magnification at that part of the scale.  Again, in practice, you will build lookup tables for this mapping.

The most proper choices in the above considerations are the ones that give you the visual results you desire.  You are not going to be making calculations based on the display values so there is not problem with "distorting" the values (i.e. non-linear mappings) to give better visual results.

There actually is a physical significance of the values of the DFT coefficients.  Discovering this was a major aha moment in my understanding of the DFT.  A DFT bin value is actually a weighted average of a set of Roots of Unity, where the signal values are the weights and they are "wrapped" around the unit circle.  The bin index is the step size in roots, and thus also then number of times the function is stretched and wrapped around the circle.  For instance, suppose you have a signal with five bumps in it.  When is stretched five times and wrapped around the circle five times, all the bumps align and throw the center of mass in that direction.  On the other hand, when it is strectched by a factor of four, and wrapped around four times, the bumps are evenly spread and cancel each other out.  This is how a DFT "plucks" out the frequencies that are in a signal.

I have written a blog article "DFT Graphical Interpretation: Centroids of Weighted Roots of Unity" to explain this.

This is long enough, hope it helps some one.

Ced