An example could be spatial abilities. There is quite a bit of between-person variability in spatial abilities. In other words, people differ in how well they solve spatial problems. The variance is a measure of such interindividual differences.

We could now be interested in, for example, whether there are gender differences in spatial abilities. If men and women differed with regard to their average spatial performance, we could say that the factor gender "explains" or "accounts for" a certain portion of the variance/variability in spatial abilities. The percentage of "explained" variance is given by squared correlation coefficients such as R squared (in regression) and Eta squared (in ANOVA).

I think the "explained by" description is bad. A variable lacks the cognitive capacity for such a task. I think saying, for example, shared variance, makes this notion easier to understand.

in the context of the general linear model, such attribution of variance does not mean very much and can be very misleading. Much better to concentrate on the interpretation of the estimated coefficients and their uncertainty. In relation to @Christians example, I would be much more interested in the sign and size of the difference between men and women and the difference in heterogeneity (so I am interested in comparative variance).

I agree with Daniel Wright that the "explained by" description can be confusing--except perhaps in true experiments where the values of a manipulated independent variable X are experimental conditions and where variations in X can be clearly identified as a cause of variation in a dependent variable Y.