"Moderator analyses revealed significant research design effects..."

What that means is that the design and sample selection of the studies influenced the outcomes of the studies to a large degree. Oftentimes, we see this happen in studies that draw from clusters of the population. Clusters are not truly random samplings. So when you take a sample population from a cluster -- be it geographic, ethnic, national, political, etc. -- you're run the risk of not taking a genuinely population-representative sample. Your sample population is likely to be more homogenous than a truly random sampling would be, as seen in a lower variance rate among respondents than you'd expect to find from a random sampling.

This is all well and good if what you're testing is the effect of X or Y on that cluster. But if you're trying to study effects of X or Y on the general population, then cluster sampling is far from ideal.

Let's say I'm designing a study called "National Attitudes Toward Ohio State Football." And for my sample population, I select 500 study participants from the state of Michigan. Would anyone in his right mind believe that my study will produce a nationally representative outcome? Similarly, if I'm doing a study called "The Effect of Gas Prices on Grocery Budgets," I might be pre-biasing my outcome if I select from a cluster of long-commute suburbanites. Or from a cluster of poor people. Or a cluster of rich people. Etc.

It's hard to know what sort of design effects were noticed in the various studies aggregated into this particular meta analysis, or how the meta-analysis attempted to compensate for such effects. But the presence of that line in the abstract gives cause for concern.