This actually appears to be an analysis of deviance table, but the principle is the same, and people still call it an 'ANalysis Of VAriance' table. The table presents information about a series of sequentially nested model fits. In the first row is the null model (without any of the variables included). Each subsequent row adds another variable to the model and information about the changes is given. It is more typical to move in the opposite order (i.e., from the 'full' model on down, dropping one variable at a time), but this is inconsequential.
The columns might make more sense if they were presented in a different order. The fourth column contains a measure of goodness of fit ($-2\times\log\ {\rm likelihood}$); bear in mind that lower values imply a better fit and that the fit has to improve upon adding a variable whether that variable is relevant or not. The second column (Deviance) is displaying the difference between that model's -2*LL and the previous model's. Column three (Resid.Df) says how many residual degrees of freedom each model has. In the first column, you see listed the degrees of freedom associated with each variable, it is the difference between that model's residual degrees of freedom and the previous model's. The thing to realize here is that the difference between two nested models' -2*LL (i.e., the deviance), is distributed as a chi-squared variable with the degrees of freedom equal to the difference between the two models' residual degrees of freedoms. The probability of seeing a difference in -2*LL that large or larger, given the addition of a variable with that many degrees of freedom is displayed in the last column (Pr(>Chi)). Thus, having stipulated an $\alpha$ / type I error rate you feel you can live with, we can see if the improvement in model fit upon adding a variable is greater than we would expect by chance alone.
