According to The British Archaeology Magazine, Neanderthals were the first species found to bury the dead. They even made smaller graves to bury their infants. As to why the practice started, It could be because of two reasons. One would be as they started settling down togather etc, a dead body would start to rot and make the place uninhabitable, so they could be burying the bodies to be able to continue living at a given location. Second would have been to just let go of someone close to you, as burying, even today is seen as a symbolism of moving on, laying them and their memories to rest etc.