I hate that it is explained that way, but reading the whole Bible gives you a better understanding of the situation. Basically, they ate something that they shouldn't have eaten and it seemingly messed up the genetic code and it was passed down through generations. Sin is what was passed down and from that, we get all these disorders and diseases. So, by the New Testament when Jesus is healing people, he is curing sin, which was the root cause for every ailment. That is why in one passage, he says your sins are forgiven and the man was healed.
Basically, sin is far more than just wrong doing, it is a defect/sickness, which is why we ultimately die as a result of it.