One approach is to consider the 95% CI as a more informative version of the p-value for the interaction. It tells you not only whether the interaction is significantly different from 0 but also indicates, in an approximate way, the power of that test; it tells you how precise or imprecise your estimate of the interaction parameter is.

Another approach is to compare the size of the interaction term (interaction parameter*time value*treatment indicator value) to the predicted Y value at several points and reason about its influence that way.

This is just a start, others will likely add detail to it.

What I meant is how to interpret the 95% confidence interval of the groups when there is a significant time*treatment interaction in the repeated measure ANOVA analysis. There are two different interpretations among my colleagues. 1. The 95%CI of both intervention and control groups should not be overlapping each others. 2. The mean of one group should not in the confidence interval of the other group. Which one is correct?

I believe both interpretations are incorrect - the first criterion is a little too strong and the second is too weak. If you question is about the statistical significance of the test then you really do need a p-value for the test of equality of the two groups unless 1) the CI's don't overlap at all (then they are clearly different) or they overlap so substantially that no one wants to claim there is a significant difference.

David is right. Your model is estimating the difference between the means and its standard error. They are trying to draw conclusions based on the individual means and their individual standard errors. If it were that simple, you wouldn't need ANOVA.