A negative covariance, or ``anti-correlation'', between two variables, just means that when one variable's value goes up, the other's tends to go down (the example I gave in class a few weeks ago was the anti-correlation between weight and jockey speed: the more you weigh, the slower your horse will tend to run). You can see how this works from the definition of covariance: . If whenever is larger than , tends to be smaller than , and vice versa, the average of the product of deviations will be negative.
Actually covariances are connected to cancellation of systematics, but not necessarily by anti-correlation: positive correlation can help too. For example, if you are trying to measure a small signal on a large background with systematic uncertainties, taking a ratio with another measurement that has correlated systematics will introduce a covariance term that helps reduce the effect of those systematics.