It was originally believed the Neanderthals were killed by humans that migrated out of Africa. However, recent evidence from genetic studies prove humans and Neanderthals were interbreeding.
Researchers at National Geographic discovered, “the Neanderthal linage disappeared, because it was absorbed into the much larger human population.” They found Neanderthal DNA in humans, especially in those with a strong European heritage.
Basically, the Neanderthals intermingled with humans until they were breed out of existence.
Source: Genographic Project: nationalgeogrpahic.com