I personally think "Birth By Sleep" Refers to the Kingdom Hearts afterlife A.K.A Rebirth/Reincarnattion. I think when a person dies, their body dissapears and the heart returns to KH. While in KH the heart falls into a deep sleep where the memories of it's passed life are forgotten. At some point in the future the heart awakens and is reborn into a new person.
Therefore in my opinion "Birth By Sleep" means a heart returns to KH, falls to sleep, later on said heart is reborn from that sleep into a new person. Thus "Birth By Sleep"
Now what does Ansem mean when he says "So many are still waiting for their Birth By Sleep" I think he's refering specificaly to Roxas, Xion, and Namine. I think Ansem discovered a way for them to get one of these sleeping hearts from KH and thus make these hearts their own, in this way those 3 are reborn. So in KH3 Sora will use KH to make 3 sleeping hearts be reborn in Roxas, Xion, and Namine and those 3 will be reborn as new people with these hearts and can live new lives as normal people.