As @Glen_b noted, F is a ratio of variances. If there's an advantage to interpreting it instead of p, it's that you don't need to convert it to F to obtain that ratio, if that's what you want to interpret. This is a somewhat circular answer, but since both are functions of one another, there's not much more to say.
One other issue to be aware of is the controversy regarding the Neyman–Pearson framework for interpreting significance statistics. There are alternatives to dichotomizing p values as less or greater than $\alpha$ for the sake of interpretation. IMO, one should have a better reason for not rejecting the null if
p = .051 (e.g., pragmatic, "real-world" costs or risks, or the ability to replicate), unless one really isn't all that interested in the alternative hypothesis in the first place. Many studies are more about effect size than about statistical significance anyway, so it would often be better to focus on effect sizes and present confidence intervals than to focus on null hypotheses, especially when statistical power is not the limiting factor in the study. For more on this, see
"Is the exact value of a 'p-value' meaningless?" For a simple repeated measures ANOVA, consider interpreting $\eta^2$ (see here for an intro), especially if you have plenty of data.