Assumption of a generation of the dataset by a probability distribution
The main takeaway from this paragraph is that the authors assume that the training data and test data are sampled i.i.d. from the same distribution, because then if we are able to estimate some statistic of the distribution we sample the training data from, we will be able to do the same on the test data.

As for your specific points:


  1) How probability distribution is generating a dataset?


We sample from a large set according to some distribution. The precise method depends on the problem.


  2) Are the generation process and probability distribution the same?


No, the generation process provides a sample from a distribution. A probability distribution is an abstract mathematical object, while a generation process is a specific algorithm that provides a dataset. In a way, the distribution provides a probabilistic description of what the generation process should return.


  3) What is the sample space and random experiment for the underlying probability distribution?


The sample space depends on the problem in question. I'm not sure what you mean by 'random experiment', but the definition of the random distribution is problem-specific.