As to your question "why did the university [at Göttingen] become so prevalent":

The answer is surely "Felix Klein". A dynamic, versatile, and brilliant mathematician, he was also a talented organizer and administrator. The list of first-rate mathematicians he brought to Göttingen is probably too long to fit within the allowable limits of a SE *answer*; some details can be found [at wiki](https://en.wikipedia.org/wiki/Felix_Klein). As editor he similarly transformed *Mathematische Annalen* into a leading journal.