Variance measures the amount of information in the data set. How?
Information is a slippery concept, so it pays to be a little concrete. So specialize to the case of a regression model. You have some variables $x_1, x_2, \dotsc, x_p$, say, which you wants to use to predict or explain $Y$. Lets say now that all the observations of $x_1$ are equal, so its variance is zero. This variable will not be useful for explaining $Y$. If you want to study, say, the relationship between education level and income, but your sample only includes college graduates without any post-graduate education, how useful will that be? It is in this sense that variance can be a measure of information in the data.
But this will only make sense in relationship to some given model. "Information" in the abstract, without any qualification, does not make sense. To see that more clearly, another example: let’s say your observations are repeated measurements of the same specimen, length, weight, whatever. The ideal case is that all are equal — no measurement error. Variance in this case represents measurement error, and higher variance corresponds to less information!