I would like to know an intuitive way to understand a Cauchy sequence and the Cauchy criterion.

The rough intuition is that if we go far enough along the sequence we get to a point where it doesn't vary very much. And if that is the case it must stay within a narrow range of values.

If we can reduce the variation arbitrarily (choose $\epsilon$ as small as we like) by going far enough ($N$ terms), then we can narrow the range as much as we like, so that there is ultimately a single value - the limit.

The value of the criterion is that it proves there is a limit without needing to know what the limit is - just using the internal properties of the sequence.

Since you asked specifically how to understand Cauchy sequences "intuitively" (rather than how to do $\epsilon,\delta$ proofs with them), I would say that the best way to understand them is as Cauchy himself might have understood them. Namely, for all infinite indices $n$ and $m$, the difference $p_n-p_m$ is infinitesmal. Such formalisations exist, for instance, in the context of the hyperreal extension of the field of real numbers.

As far as the particular series you asked about, what is going on is that the book is considering the sequence $p_n$ of partial sums of the series, and applying the Cauchy criterion to this sequence. Then the difference $p_n-p_m$ is the expression $\sum_m^n$ that you wrote down (up to a slight shift in index).

Some thoughts on Cauchy can be found here.