What does the evidence suggest about how Neanderthals became extinct?
This is one of those active areas of research where there isn't a lot of consensus among researchers. There isn't even consensus as to whether it was an extinction in the strictest sense, because we now have evidence from genomic research that Neanderthals interbred with modern humans. Research is still being done on a lot of different hypotheses.

This article by K. Harvati is a decent short summary of the current state of the field, and includes references to recent articles.