FMCW Radar Working Logic Question
Grace Hopper used to use pieces of optical fibre, to show how "long" is a nanosecond at the speed of light. It is approximately 30cm and a nanosecond is a sort of a time interval that can be easily measured, accurately, without expensive lab equipment (e.g. with a simple counter, using a crystal as its time basis).

So, the "speed of light" is not something that is inaccessible or somewhat of a big problem for electronics. Especially at Radio Frequencies (RF), the speed of light is even a "necessity" if you are trying to observe phase shifts that imply extremely small distances in space.

The point about the continuous wave radar is that it can use its local oscillator for the "demodulation" of its transmitting wave. And using different patterns of modulation, it can return different types of information (for example range and / or velocity of a target). 

So, at its heart, the Continuous Wave radar is based on measuring phase shift via a phase detector. And again, this operation, at its heart, is extremely simple: Multiply two sinusoids at frequencies $f_1,f_2$, you get a signal with two components, one at $f_1+f_2$ and one at $f_1-f_2$. At the special case where $f_1=f_2$, you get a component at $2 \cdot f_1$ and a component at 0 Hz, or a DC component, a constant value added to the signal. If, in addition, you vary the phase between $f_1,f_2$ (while still keeping $f_1=f_2$), then the DC component is proportional to the phase difference.

This last point is also the basic principle by which Phase Locked Loops produce the error signal that they use to tune their local oscillators to the frequency and phase of an incoming signal.

So, having said (and hopefully understood) all this, let's look at what you ask:


  Basically how IF shift in signal occuring. 


The shift is occuring because of phase shift. Another way of viewing phase ($\phi$) is as the first derivative of frequency ($f$). Therefore, if you integrate phase, you get frequency. In other words, many many many little phase changes, accumulated in the right way ammount to whole changes in frequency. For an explanation of why does the shift occur in the physical world and within different propagation media, please see the Doppler Effect.


  Is it occuring for every frequency part of the chirp signal (6 GHz for 500 us, 6.01 for 500 us ... up to 6.1 GGz)... 


Yes. The same phenomena that apply at some $f$, also apply for some $f + \Delta f$, where $ \Delta f$ is some frequency offset. So, whether looking at a single sinusoid or a set of sinusoids traveling together, there will be phase changes observed, proportional to each frequency's wavelength. Which brings us to the second part of this question:


  ...or is it occuring as a result of whole chirp package?


Again, yes, in the case of the Continuous Wave radar, as it was mentioned before, you use the local oscillator to demomdulate the incoming (reflected) wave. Therefore, your "window of integration" (the window within which you assess phase change) would have to be proportional to the "length of the pulse".

Very briefly: If you get very early reflections, at timings that would cause a negligible phase change between the transmitted component and its reflection then that would register a phase change of zero and consequently also a default distance. In other words, the frequency and pulse timing of the CW radar set limits to the distances it can observe. Any reflection by targets at LESS than that critical distance will not be registered and any reflection by targets at GREATER than that critical distance will also not be registered (or contribute to noise if they happen to be received within a subsequent "window"). It is reflections within specific limits that cause detectable phase changes which can then be used to infer either range (distance to target) and / or velocity of target with respect to that of the source (if they are both moving).

For more information, please see differences between "types" of CW, purely by changing the "pulse" pattern, via this link.

Hope this helps.