Can anyone explain how does CZT (Chirp Z Transform) really help in 'spectral zooming'?
The CZT allows for a fairly general evaluation of the Z transform - the more general evaluation path looks like a spiral, so it has a radial component step size as well an angular step size.For  spectral zooming, you're only using a subset of this. You're evaluating around the unit circle and only for a small set of frequencies.

The Zoom FFT can be implement in a more traditional manner:


Frequency shift by multiplying time domain signal by a complex exponential - baseband around the center frequency.
Low-pass filter and decimate - to reduce the sampling frequency.
Apply a (shorter) FFT


Note that the FFT in step 3 is shorter because of the reduced sampling rate. To get a frequency resolution of $\nabla f$ you need a time series of length $T=1/\nabla f$ and at a lower sampling frequency you can get that time length signal in fewer samples. This series of steps applies a series of operations to the signal.

Now another way of looking at those series of steps is to apply them to the filter rather than the signal. Now you're shifting a low-pass filter in frequency so it becomes a complex band-pass filter and the decimation step causes duplicates of your band-pass signal to appear around 0 Hz (and throughout the entire frequency spectrum). At this point you can apply a shorter FFT. To make the two series of steps exactly equivalent there are some restrictions on the sampling frequency - If you shift the filter to a center frequency of $f_c$ then you have to be careful of the decimation factor if you want the frequency content at $f_c$ to be mapped exactly to 0 Hz.

The CZT for Zoom FFT is essentially doing these same steps. If you examine the CZT from this point of view you can work the mathematical equivalence.