I'm not sure (particularly as someone who was raised by parents who were biological parents) that I can come up with one specific "most important thing" in reply to this question. I think, so often, it may depend on the individual birth parents and individual child and circumstances and history of all involved.
As someone who knows close to nothing about a lot of details of my own grandparents' lives/histories, I don't happen to believe that a lot of details about family background (particularly past the birth parents) are something we can't live happily enough without.
I think all adopted children need to know that if their birth mother had been able and/or ready to take on the responsibilities/role of being "the kind of mother a child needs" she would not have chosen to place her child for adoption (or else had him/her removed from her care by the state). I think, even in the case of a girl or woman getting pregnant by mistake and saying, "I just don't want this baby," an adopted child needs to know his/her birth mother was (by virtue of risking an accidental pregnancy to take place at all) not mature enough (no matter how old she was) to be good and sure she didn't have a pregnancy if she knew she wouldn't want the baby. Then, too, there are girls/women who are the victims of rape and/or incest and either can't or shouldn't be expected to raise a child after herself being emotionally harmed to the point beyond her ability to do a good job of it, or love the child.
I guess what I think all adopted people need to know is that birth mothers aren't monsters who didn't want their baby just because they didn't want him/her. The most careless or indifferent of them have been, in one way or another, damaged to the point of being careless of indifferent. Many are victims themselves in one way or another, and the choice to place a child for adoption is pretty much always a choice aimed at at least trying to offer the child the kind of parent(s) every child deserves and needs.
Other than that, I think they need to understand that no matter who/what their birth parents were; when babies are adopted in infancy their personality, health, and so many other things about them is formed as a result of how they're nurtured by their adoptive parents; so sometimes it doesn't really matter much who or what the birth parents were nearly as much as many people believe (and lead adoptive children to believe).