As for your question regarding Why?
As far as I understand this is to be able to get more academic exposure to different ideas. One could imagine that someone who trained for around 8-10 years in the same place might leave with some bias from working with the same people and being exposed to the same ideas for that period of time.
I completed my BS and MS at the same university and then traded institutions for my PhD. This wasn't as much for academic exposure as finding the best program for me. However, having made the switch I have appreciated the different environment and meeting new people. I have only positive things to say about going to a different institution for more advanced degrees.
This advice is most relevant when considering institutions for a PhD. Someone who is pursuing a terminal masters degree probably doesn't need to worry much about going to a different institution.
Of course you will find many people who agree and disagree with this rule. I know several people who got all their degrees from the same institution and are doing just fine and others who decided to go to a different institution. The bottom line is you need to find the program that is going to fit you best.