Good question! It was a while since I read Pearl, so I had to go back to my notes. I'm not sure if my explanation will be satisfactory to you, but I'll try.
First of all, it's very important to understand what P(Y|X) means. It could mean that X increases the propensity for ("cause") Y directly. This is the case where e.g. X is "speeding in a residential area" and Y is "pedestrian killed".
However, P(Y|X) could also mean that something else increases the propensity of both X and Y. In my country, we could have Y being "melanoma" and X being "high vacation spending". This is not because high vacation spending directly cause melanoma, it's because people who fly to sunny destinations every year both expose themselves to sun and spend a lot of money on vacations.
The way to interpret P(Y|X) is not that X causes Y, but rather that X is evidence of a situation in which Y is more likely.
With that out of the way, we can move on to your actual question. In the equation you wrote down, Z is that "situation in which Y is more likely", the one that X is evidence for.
More concretely, Z could be "vacations in the sun".
To find out if high travel costs truly cause melanoma directly, we have to control for "vacations in the sun". The experimental way to do this is to round up a bunch of random people, force half of them to spend a lot on their vacations (without changing anything else about their vacation plans) and then seeing if those people develop melanoma more often (then it would be just from spending.)
That experiment is what Judea would write down as P(Y|do(X)). Ensure people spend, see how likely melanoma is then.
We can't do that experiment, obviously.
But we can simulate the experiment from observational data, if our causal model is complete enough! That's a really neat trick.
In the sum
$$\sum_Z P(Y|X,Z) P(Z)$$
Two things are happening: first, we are looking at the probability of Y given that we are informed about both X and Z. In the example, this means that someone asks us to predict a person's risk of melanoma with information on their vacation spending and how often they vacation in the sun. We intuitively know the outcome here: since spending doesn't cause melanoma, we can think of this as predicting melanoma given just the information about vacations in the sun. (Then we multiply by the probability of spending vacations in the sun to get the proper joint probability.)
A more interesting example could be where e.g. Y is bad health, and X is low income. We know that low income causes bad health, we know that bad health causes low income, and we know that a lot of things can cause both bad health and low income.
Importantly, P(Y|X) means "if we observe someone with low income, what's their probability of bad health?" Note that this says nothing about low income causing bad health. It's saying that low income is evidence of a situation and personal history that can also lead to bad health.
There are many such things: war, drought, etc, etc.
The meaning of P(Y|do(X)) in that context is "if we force people to have bad income, what happens to their risk of bad health?"
This eliminates all other variables (war, drought, etc) from the equation and makes it all about the effect of income on health.
Observationally, we look instead at the sum:
$$P(health|income, war) P(war) + ...$$
Meaning "what would we predict about someone's health given that we know they have low income and have suffered from war? And by the way, how many people suffer from war, more generally?"
When we do the same for all the hidden factors, we have accounted for them separately and they no longer influence the probability of bad health any more than they do in the baseline population. Thus any effect we see is purely from low income.
