The point depends on differences between confidence intervals and credible intervals.

Roughly, frequentist confidence intervals, but not Bayesian credible intervals, have the following coverage guarantee: if you repeat the sampling and analysis procedure over and over, in the long-run, the confidence intervals produced cover the truth some percentage of the time corresponding to the confidence level. If I set a 95% confidence level, then in the limit, 95% of the intervals I generate will cover the truth.

Bayesian credible intervals, on the other hand, tell us what we believe (or should believe) the truth is given the data. A 95% credible interval contains 95% of the probability in the posterior distribution (and usually is centered around a point estimate). As Gelman points out, Bayesians can also get a kind of frequentist-style coverage by averaging over the prior. But in Wasserman's cartoon, the target is a hard-core personalist who thinks that probabilities just are degrees of belief. No averaging is done, because the credible intervals are just supposed to represent the beliefs of that particular individual. In such a case, we have no guarantee that the credible interval covers the truth even occasionally, even in the long-run.

Take a look here for several good explanations of the difference between confidence intervals and credible intervals that are much more detailed than my comment here.