Answer:
It goes all the way back to the Garden of Eden in the book of Genesis. God did not want to create robots that walked around saying, "I love you, God." Instead, God wanted free-will love. He gave humans ultimate freedom and happiness except for one thing: the fruit from the tree of the knowledge of good and evil. This was the only rule Adam and Eve had to follow. Everything else was theirs for the taking and enjoyment. God promised Adam that if he broke this one rule and ate from the forbidden tree, then Adam would die that same day. Adam would prove his love for God by being obedient.
Now, we know from Genesis that Adam did not physically die that same day. But on that day when he ate the fruit, Adam was making a choice to cut off his unlimited, perfectly happy access to God. The death was spiritual. This is what John means by all men are spiritually dead. We inherited from Adam not just the physical characteristics of a human being, but the spiritual as well. We are separated from God. The only One who can restore that relationship is God Himself. He chose to do so by demonstrating how much He loves us by giving up His one and only Son, Jesus, to die in our place. In taking the sin of the world on Himself, Jesus died the same death Adam did -- separation from God, spiritual death. In shedding His blood for us, Jesus also died physically.
The implications of "spiritually dead" are numerous, but a few stand out. First, we cannot gain God's favor by anything we do. Physically dead people can't do anything physical. Spiritually dead people can't do anything spiritual. Therefore, we cannot achieve a new relationship with God by anything good that we do.
Second, as already mentioned, our fellowship with God is destroyed. Only He can restore that communion if we ask Him to forgive us through what Jesus has done for us on the cross and from the grave (rising from the dead, spiritually and physically).