Why did it not keep going? What made it diee out and allow Europeans to recover the population numbers?
It didn't. The strain of Y. pestis that caused the Black Death in the 14th century is very similar to modern strains. So similar, in fact, that we now have begun to postulate environmental causes for the massive death rate. What caused it to end was that there weren't many people left who hadn't got it and weren't dead.
Some reading:
S. Haensch, B. Bramanti, R. Bianucci, M. Signoli, M. Rajerison, M. Schultz, S. Kacki, et al. “Distinct Clones of Yersinia Pestis Caused the Black Death.” Public Library of Science Pathogens 6, no. 10 (2010).
Benedictow, Ole Jørgen. The Black Death, 1346-1353: The Complete History. Woodbridge, Suffolk, UK ; Rochester, NY: Boydell Press, 2004.