Why does a cell produce a lot of p53 protein only to later degrade it?
This answer is not reserved for the example you cite but for all proteins/RNAs which have high turnover. So your question generalizes to 

What is the advantage of having high turnover? or in other words, What is the advantage of regulation via degradation?


Fast responsiveness. If your protein/RNA has a low turnover then it would take a lot of time for it to clear up from the cytoplasm even after the activating signal is gone. Enhancing the degradation will make the response fast. Cell cycle proteins such as p53 have to clear up as soon as the cell progresses to the next phase.
In certain cases an additional degradation control makes the regulation more efficient. If the "regulated" gene happens to be in turn an activator of many downstream genes, then efficient control actually helps in saving the cellular resources. I can't cite any literature for it but I have observed this in one of my unpublished research work using mathematical models.