Can anyone give a real life example to illustrate why does the principal axis that has the maximal variance retain the most information? One job in PCA is to maximize variance, because the 
 principal axis that has the maximal variance retain the most information.
Why is that? How to understand this in a easy or concrete (such as a real life example) way?

This could be quite difficult to be seen in a real life example since it is a theoretical or abstract assumption.
When you do PCA you want to project data in a low dimensional space. This projection obviously will loss information so you want to retain the most information. As the PCA assumption, the eigen-values of the matrices will show the variance retained. 
This variance will illustrate the error reconstruction of data, you want to have a rich representation in the low  dimensional space, as good as, you can return to the high dimensional losing little information as possible.
Think, if you go to a space with few variance you can think you are projecting onto a point. How do you return to the other space? The most variance in data the better you can reconstruct data.
The PCA can be seen as maximizing this variance, so the most variance the best is the fit done. If the variance is near to 1, it is said that the fit is very good and data relies on the space you are projecting.
You have here a good lecture on this: https://www.stat.cmu.eitu/~cshalizi/350/lectures/10/lecture-10.pdf
