This seems pretty unsurprising to me.
In your model, the ICC can be interpreted as the proportion of the variance not explained by covariates that is due to variation between classes.
If instead of using "proportion of fully prejudiced respondents in the class", you used "average level of prejudice within each class" as a covariate, this would certainly reduce the ICC to zero, as this covariate explains all the variation between classes by definition.
So if "proportion of fully prejudiced respondents in the class" is strongly correlated with "average level of prejudice within each class", as you might well expect, it's not surprising that it also reduces the ICC to zero.
The pseudo-$R^2$ value that is reported is computed with: $$R^2 = \frac{\hat{\tau}^2_{RE} - \hat{\tau}^2_{ME}}{\hat{\tau}^2_{RE}},$$ where $\hat{\tau}^2_{RE}$ is the (total) amount of heterogeneity as estimated based on a random-effects model and $\hat{\tau}^2_{ME}$ is the amount of (residual) heterogeneity as estimated based on the mixed-effects meta-regression model. Note that this isn't anything specific to the metafor package -- it's how this value is typically computed in mixed-effects meta-regression models.
This value estimates the amount of heterogeneity that is accounted for by the moderators/covariates included in the meta-regression model (i.e., it is the proportional reduction in the amount of heterogeneity after including moderators/covariates in the model). Note that it does not involve sampling variability at all. Hence, it is quite possible to get very large $R^2$ values, even when there are still discrepancies between the regression line and the observed effect sizes (when those discrepancies are not much larger than what one would expect based on sampling variability alone). In fact, when $\hat{\tau}^2_{ME} = 0$ (which can certainly happen), then $R^2 = 1$ -- but this doesn't imply that the points all fall on the regression line (the residuals are just not larger than expected based on sampling variability).
Regardless, it is important to realize that this pseudo-$R^2$ statistic is not very trustworthy unless the number of studies is large. See, for example, this article:
López-López, J. A., Marín-Martínez, F., Sánchez-Meca, J., Van den Noortgate, W., & Viechtbauer, W. (2014). Estimation of the predictive power of the model in mixed-effects meta-regression: A simulation study. British Journal of Mathematical and Statistical Psychology, 67(1), 30–48.
In essence, I wouldn't place too much trust in the actual value unless you have at least 30 studies (but don't quote me exactly on that figure). For a nice exercise, you could use bootstrapping to obtain an approximate CI for $R^2$. Pretty much all you need to know to do this is explained here:
http://www.metafor-project.org/doku.php/tips:bootstrapping_with_ma
Just change the value that is returned by the boot.func() function to res$R2 (and since there is no variance estimate for $R^2$, you cannot get the studentized intervals). In your case, you will probably end up with a very wide CI (possibly extending pretty much from 0 to 100%).
Best Answer
The ICC in the example/model you linked to (and the correct equation is $ICC = \frac{\sigma^2_1}{\sigma^2_1 + \sigma^2_2}$, that is, we need to plug the variances into the equation, not the SDs) represents the correlation between the underlying true effects from the same cluster.
The 'between-cluster' $I^2$ is indeed quite similar and is given by $I^2 = \frac{\sigma^2_1}{\sigma^2_1 + \sigma^2_2 + \tilde{v}}$, where $\tilde{v}$ can be interpreted as the 'typical' sampling variance. This can be interpreted as the amount of total variance (the sum of the between- and within-cluster heterogeneity and the amount of sampling variability) that is due to between-cluster heterogeneity. One could also interpret this as the correlation between two observed effects from the same cluster (assuming that these two observed effects have sampling variances equal to $\tilde{v}$).
The 'within-cluster' $I^2$ is $I^2 = \frac{\sigma^2_2}{\sigma^2_1 + \sigma^2_2 + \tilde{v}}$. It indicates how much of the total variance is due to within-cluster heterogeneity. This one doesn't lend itself to a correlation interpretation.