"Slight bias" doesn't have a specific meaning, but what the authors are doing is talking about the magnitude and direction of their effect measure, and decoupling that from it's significance.
Their results show a slight effect, and that alone may be interesting, even without significance - if it agrees with other studies, what they mean by slight, etc.
Also, keep in mind that, while we treat it as binary, "not significant" is not an on-off switch. It's probably not equally likely that the effect is in the other direction, and in some cases of non-significance (p = 0.0506) it's actually quite unlikely, but still non-significant.
To use two other examples from research I've seen:
The association between many environmental exposures and rare disease outcomes may never be significant - the outcomes are too rare, the effects small but non-zero, the sample sizes too small, and the studies simply too expensive to run. So you may have twenty non-significicant results, but if they're all in a single direction, that's information.
In a study I conducted, the hazard ratio of a study turned out to be 1.97 with a confidence interval from 0.96 to 4.01. Not significant, but my sample size was dictated by the size of a particular data set, not the power needed to see an effect of that size. And the vast bulk of the evidence is that there's a positive effect - while it is possible that the effect is zero or a little below, the most likely estimate is above zero, and probably still worth reporting.