Why does a F value of 1.00 have a p-value which is less than 1?
The F distribution is commonly used to assess the variance associated with groups and the variance associated within the groups.
$$ \mbox{Expected F = } \frac{\mbox{true variance within groups + variance between group means}}{\mbox{true variance within groups}} $$
If the group means are equal the above ratio will be exactly equal to one.
However in general we only sample a certain number of people which give rise to the variances and F ratios above.
If we sample only a handful of people we will need a bigger observed F ratio (>1) to convince us that the group means differ than if we have a large total sample size. So the critical values which determine just how big a F ratio we need to suggest groups differ need to take into account sample sizes.
The more groups (which are relate to the first degree of freedom (df1) in a F ratio) and larger sample size (which is related to df2) we have the less the F ratio needs to deviate from unity to reject the null hypothesis of a unit F ratio and, therefore, suggest there are group differences.