What does it mean for a statistical test to be "robust"? Is there an intuitive way of understanding what these two sentences mean and why they're true?:
"ANOVA is 'robust' to deviations from normality with large samples", and...
"ANOVA is 'robust' to heteroscedasticity if the groups have similar sample sizes".

When we say that a procedure is "robust" or "robust to [a particular failure of assumption]" we mean that the procedure still works well when the underlying assumption is not met.  So, in the present case, the quoted statement is telling you that, under the stipulated conditions, ANOVA still works well even when the normality or homoskedasticity conditions in a model are not a realistic reflection of the data.
