I've noticed that the compressor in refrigerators and A/Cs shaking during turning off.
Why this is happens?

I would like to preface this statement by saying that I am not a physics professor, but I think I know why.

If you have an object (like a compressor) mounted on springs, it will have a very noticeable resonant frequency. If you kick it, it will jiggle around at a specific frequency for a bit before settling down. When the compressor is running, the movement inside the compressor is not at the resonant frequency of the compressor mounted on springs, but when it slows down (like it does after turning off), for a moment the internals will be moving at the resonant frequency of the compressor. This will cause it to jiggle a lot for a little bit.

The biggest part is exactly that.
For the springs to really do their job and that is to reduce vibration transfer from the core to the shell, the resonance frequency must be way below the vibration frequency (related to the spinning assembly) you want to reduce. Now as it spools down when switched off, the rotation speed needs to pass the spot, where the vibrating frequency matches the resonance and so excites a lot of amplitude in the resonating system (the springs with the mass) and this large amplitude is what indeed cause the shake. It can also go as far as the parts really physically hitting each other (probably there will be "bumper" protrusions on the parts explicitely designed to take these hits), causing the really loud rattle some compressors tend to make. The same happens at startup, but there the rotation speed tends to change way more rapidly, so it is in the resonance area for way shorter time, so excites lower amplitude.

Another mechanism could be, when the thing uses resonant vibration dampers. These are added mass/spring combinations with their resonance frequency designed to match the vibration frequency. That way they are "sucking" the vibration energy from the compressor body, so eliminating its amplitude without the need to transfer the forces (so the noise,...) to the outer shell and the equipment body itself. Here once the compressor starts to slow down, it starts to generate vibrations out of the resonance of the dampers, so the dampers stop eliminating it, so it turns into extra compressor shake.
These resonant damper will be used only on fixed speed (because the resonance can not be changed) compressors explicitely designed for quiet operation (so there is financial budget for these things to be made). It is one of the tricks which are able to run a high efficiency wide pressure range piston compressor in an AC/heat pump combo unit really quiet, but it does not allow variable speed regulation.

@Medved: Are the purpose of the springs to absorb shocks and to make the compressor quiet?
I've seen this phenomenon also with old local made A/C from the 90's which had relatively noisy compressor. Some of them even had rotary vane compressors or scroll compressors.

All of that is to reduce the compressor noise. And before that it was to reduce the vibrations transferred to the mounts and frame, so it won't fall apart (but that level was 100 years ago).

Rotary compressors are inherently quieter, but their drawback is their geometry must match the compression ratio during operation. Any variation in operating conditions and the compressors become very inefficient.
So if you have exactly known refrigerant, exactly known temperatures and power level, you may design/select the vane/scroll compressor with matching compression ratio for that and it will operate well. But in real life the temperatures vary quite a lot (mainly with reversible heating/cooling heatpumps), plus the exact parameters of the systems are hard to predict during design, so the efficiency suffers a lot (the compression pocket opens at high pressure imbalance, leading to restricted crossflows so useless gas heating so losses).

The reed valves on piston compressors have the huge advantage to opening and closing exactly at point with nearly none pressure difference, so make the compression ratio adaptable to what the actual conditions need. So although a bit less efficient than the csroll/vane at their optimum condition, they do not get worse once the conditions vary (AC/heatpump,...). The drawback is these are reciprocating machines, so inherently creating vibrations that need to be adressed. And that is, where all the suspension springs and resonant dampers and many more vibration suppression techniques come in.