I think that the only intuition behind the test statistics of those tests would be related to the p value (significant or not) - the conclusions to be drawn depend on the research question. However, if you see how such test statistics are calculated you can try to get some additional intuition: e.g., if you compare two cumulative density functions you can determine the intervals, namely values, for which the CDFs are similar and intervals, for which the CDFs diverge - this gives you additional inference, e.g. if the test statistic is not significant for complete samples but on some intervals it may still be significant. VK

I think that the only intuition behind the test statistics of those tests would be related to the p value (significant or not) - the conclusions to be drawn depend on the research question. However, if you see how such test statistics are calculated you can try to get some additional intuition: e.g., if you compare two cumulative density functions you can determine the intervals, namely values, for which the CDFs are similar and intervals, for which the CDFs diverge - this gives you additional inference, e.g. you can see for which intervals the test statistic reaches its maximum (and therefore the CDFs diverge). VK