Does a PhD application decision letter give the reasons for the decision? Why or why not?
In my experience in the US, I have never seen any sort of decision letter for an application (whether for Ph.D. or any position) that gives reasons for the decision. It may be different in other countries, but in the US at least, there are many good reasons to never give such information formally.

The actual decision-making process is never as clean and clear-cut as anybody wants it to be, and with the limited information available in applications and even interviews, there are going to be cases where people make mistakes or make good decisions for unclear or hard to state reasons.  Lots of good people may get turned down for reasons having nothing to do with them, such as just having too many good candidates in the same area.

If an institution gives real reasons in an official statement (as opposed to something meaningless and bland like "not an appropriate fit at this time"), then it is opening up the door for all sorts of potential problems and disputes, not to mention potential legal liability if something could be construed as relating to a protected category such as gender, age, religion, or ethnicity.  They also have to worry about seriously unbalanced people who may take it as a personal insult and begin some sort of stalking or harassment campaign (it happens!).  Not giving reasons may feel unsatisfying and cowardly, but it is an easy and safe path for an institution, and as such is generally official policy.

Now, if you have an informal trusted relationship with somebody who was involved in the decision process, then it is sometimes possible to get a strictly unofficial take on what may have happened.  You might learn things you didn't want to know about how sausage is made, however.