How does Genesis explain the origin of death and suffering?

Answer 1

It explains that death and suffering are the result of disobeying God (Genesis ch.3). (Note that this does not directly apply to all suffering.) See also:

More about suffering

Answer 2

A simplistic reading of the Book of Genesis suggests that Adam were not destined to die, but that God decreed they would die because they ate the forbidden fruit. However, scholars say that the story is not about Adam and Eve being punished for eating the fruit of the tree of knowledge, but that what followed was a necessary outcome.

After Adam and Eve had eaten of the Tree of Knowledge of Good and Evil they became like gods (Genesis 3:22: "And the LORD God said, Behold, the man is become as one of us..."), except that they were still mortal. They now knew that they were destined to die, but that by eating of the Tree of Life, which was also in the Garden of Eden, they would then also be immortal just like the gods.

God told Eve what she now already knew: that she would bear children in pain. And he told Adam what he now already knew: that he would one day die. Certainly God was angry, and cursed the ground. But it was so that they would not also eat of the Tree of Life, that he had no choice but to banish them and place cherubim and a flaming sword at the entrance of the Garden of Eden, to keep [block] the way to the Tree of Life (Genesis 3:24). Without the idyllic existence offered in the Garden of Eden, Adam would have to work all his days, to provide food.

Leon R. Kass (The Beginning of Wisdom: Reading Genesis) says the story of creation that begins in Genesis 2:4b should be regarded as a mythical yet realistic portrait of permanent truths about our humanity, rather than as a historical yet idealised portrait of a blissful existence we once enjoyed but lost.