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,at wiki. As editor he similarly transformed Mathematische Annalen into a leading journal.