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.

It's enough to consider the special case $\mathbb R$ for the purpose of understanding the gist of it. So, we'll assume everything happens in $\mathbb R$.

For a sequence $a_n$ to converge to a finite limit $L$, means that $$\forall \epsilon >0 \exists N s.t. \forall k>N: d(x_k,L)<\epsilon$$ which intuitively means that for any prescribed positive distance $\epsilon$, from some index onwards, all elements in the sequence are within that distance to the limit $L$.

Now, for the sequence to be Cauchy means that $$\forall \epsilon > 0\exists N s.t. \forall n,m>N:d(x_n,x_m)<\epsilon $$ which intuitively means that for any prescribed positive distance $\epsilon$ from some index onwards all elements in the sequence are no more than $\epsilon $ distance from each other.

Before we further examine the crucial differences let us remark that the oh so important property of $\mathbb R$ (and many other spaces) is that they are complete: A sequence converges if, and only if, it satisfies the Cauchy condition.

So, the conditions (in a complete space) actually mean the same thing. So, what are the differences? Well, the condition for convergence to a limit requires you to specify a limit. The Cauchy condition does not, and this is a great thing that occurs often when you want to show something converges but you have no idea what it might converge to. So you show it is Cauchy (in a complete space) and conclude it converges.

Intuitively, it is clear that any convergent sequences is Cauchy. This is so since if all elements from some index onwards are very close to $L$ then they must be very cose to each other as well. The converse though is not clear and not always true (as said above, it is the completeness property of the reals which is not a trivial matter). For instance, in $\mathbb Q$ with the standard metric, any sequence of rational numbers that converges in $\mathbb R$ to an irrational number is Cauchy but fails to converge in $\mathbb Q$.