https://ibb.co/LrBWpfD

The sampling in the first stage should have brought the aliases already, why does the author claim that it is an aliaa-free system?

S(z^M) should be a function of z^M, which cannot mitigate the aliases, for example, s(z^M)=z^M or even s(z^M)=1.

The full book is here: https://authors.library.caltech.edu/records/rmhds-22q28

The phase of the aliases rotates after each delay prior to the down-samplers, and then again after the up-samplers in each of the recreated Nyquist zones. Due to this rotation that is proportional to frequency (a delay of one sample in the time domain is a linear phase in frequency from 0 to -2pi for the frequency from DC to the sampling rate), we get perfect cancellation of all aliases except in the region where the signal originally started. It's a worthwhile exercise to try it yourself with M=2 with two tones (one that would alias) so that you can see for yourself how that works. The purpose of this is to demonstrate how poly-phase decomposition works. 

Thanks for your reply, Dan.

But I am still confused, for example,

1. What's the relationship between the down-sampling rate and the incoming signal x[n]?

2. The incoming signal x[n] is discrete-time, if the incoming signal is x(t) (continous-time), should such multirate system still be an alias-free system?

The structure your proposing wouldn't be feasible with a continuous time input, as you would also need to reconstruct to a continuous time output - and to do it with the block diagram you have would need each delay to be infinitely small (and an infinite number of them), so perhaps that is what you are struggling with?  

I expanded the answer to your same question posted on DSP.StackExchange here: https://dsp.stackexchange.com/a/93647/21048. Down-sampling and Sampling with an ADC are the same process: sampling with an ADC is down-sampling from and infinite sampling rate. 

See the similar example I provided for the down-sampling that occurs in the FFT. I recommend working through your case with similar detail and the reconstruction of the signal with no aliasing should then be clearer.