Sampling distribution of the p-value In Bayesian Data Analysis 3 (page 151) in the sub-section P-values and u-values there is a part which reads:

In the special case that the parameters $\theta$ are known (or
estimated to a very high precision) or in which the test statistic
$T(y)$ is ancillary (that is, if it depends only on observed data and
if its distribution is independent of the parameters of the model)
with a continuous distribution, the posterior predictive p-value
$\text{Pr}(T(y^{rep})>T(y) \mid y)$ has a distribution that is uniform
if the model is true. Under these conditions, p-values less than 0.1
occur 10% of the time, p-values less than 0.05 occur 5% of the time,
and so forth.
More generally, when posterior uncertainty in $\theta$ propagates to
the distribution of $T(y|\theta)$, the distribution of the p-value, if
the model is true, is more concentrated near the middle of the range:
the p-value is more likely to be near 0.5 than near 0 or 1. (To be
more precise, the sampling distribution of the p-value has been shown
to be "stochastically less variable" than uniform.)

I don't understand the second paragraph and would greatly appreciate any further explanation on this point, or an example.
I think the first paragraph I've quoted refers to the "classical" p-value distribution under the null-hypothesis(?).
Gelman also has a paper on this topic which mentions the same point as above.
 A: Suppose that we have a sample of $y$ and we are fitting the following model:
$$y\sim N(\mu, 1).$$
From this we obtain the posterior of $\mu$. Now we generate $y^{rep}$ from the same distribution: $N(\mu, 1)$. What do we expect when comparing the density of $y$ and $y^{rep}$. If the model is correctly specified, the density of $y$ should be covered by the densities of $y^{rep}$.
In the figure below

we see that $y$ is not covered very well.
The question now is that why we need statistic $T$ in your quotation. Because, the graph just only gives us a qualitative idea. We need a number, that is ppp-value. In addition, comparing two distributions is not always easy as comparing to use statistics,
For example, suppose that $y$ is sampled from a skewed distribution but we fit $y$ as a $N(\mu, 1)$. Then skewness of $y$ is far from 0 while that of $y^{rep}$ is around 0 (since it is generated from $N(\mu, 1)$ (which is symmetrical)). In this case, of course, we fit a wrong model, and we can easily see that skewness of $y$ does not locate in the typical range of the skewness computed from $y^{rep}$. In other words, ppp-value is close to 0 or 1.
