Matter is corruptible by nature. Death is simply the natural result of having a body made up of corruptible matter. Thus even animals that don’t age (which is to say, have a chance of dying at any given moment that does not increase with time) still die, because at some point something will cause their bodies to stop functioning.
In the beginning, God gave man the preternatural gift of immortality, meaning that even though we were naturally capable of dying, God would ensure that nothing would ever actually cause us to die. When Adam and Eve sinned, God simply withdrew this gift and allowed nature to take its course.
It does seem that, prior to the Flood, human beings either did not age or aged at a slower rate. According to the Book of Genesis, God did actively change the human lifespan at that time. But death is still caused naturally, God doesn’t need to specially intervene for a person to die.