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".
 A: We must be specific about what the claim is. It's not sufficient to wave our hands and say something vague like the test "works well" in those circumstances -- that is not what was examined in order to make the statement.
Both statements are specifically about accuracy of the significance level (a.k.a. "level-robustness").
That is to say, the type I error rate is claimed not to be too far from what you would calculate/choose under the (violated) assumption in those circumstances.
Even in that restricted sense, these sorts of general claims are too vague to be useful in practice, however. For example, you don't really know how large is sufficiently large for your purposes in the first case, because you don't know the population distribution (if you did, you wouldn't need to consider this issue at all!).
Of course, significance level is not the only consideration with tests. Certainly I'd hope that people care about power. Sadly, however, the direct evidence that the people who repeat these statements care much in practice is weak when common statements like these so rarely are accompanied by the merest mention of what happens with power.
In the first case, large samples don't save you when you're looking at relative efficiency (the relative sample sizes needed to achieve a given level of power) -- and relative efficiency can be arbitrarily poor in large samples -- so if your sample sizes were large because your anticipated effect size was small, you might have some potentially serious issues.
A: Roughly speaking, a test or estimator is called 'robust' if it still works reasonably well, even if some assumptions required for its theoretical development are not met in practice. Comments:

*

*If you need to do one-factor ("one-way") ANOVA for data with different variances at each level of the factor, then it is best to use some variant of one-way  ANOVA such as oneway.test in R that does not require equal variances.
As you say, a 'pooled' t test or simple one-way ANOVA where the numbers of replications per factor differ greatly, may be problematic if variances also differ among levels of the factor.


*Some texts seem to say 2-sample t test and one-way ANOVA are OK for non-normal data whenever there are more than 30 replications per group. But this may not be true if data within groups are highly skewed.


*If levels of 2-sample t or one-factor ANOVA are far from
normal, but differences between groups are mainly a 'shift'
of location (with little change in shape or variance) then
it may be best to use Welch t test or Kruskal-Wallis nonparametric test instead of t or ANOVA, respectively.
Note: I could show an example to illustrate, if you could say
what test is of particular interest and what assumption
you feel unsure of.
A: 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.
