Medicine helps keep people alive.
Summary
Medicine, also called "Health Pack" in game, has two primary functions. While in Train Mode, the item is used to keep passengers alive. If a passenger is wounded, they are bleeding, and only Medicine can keep their gauge from hitting zero. When the gauge does hit zero, the passenger immediately dies, causing you to lose any bonus they would have awarded you.
While in Station Mode, medicine heals the player character, and is a vital way of extending your longevity while exploring.