People give things up starting on Ash Wednesday in observance of the season of Lent. Lent lasts for 40 days and ends on Easter Sunday. This 40 days is in remembrance of the 40 days Jesus spent alone in the wilderness fasting before embarking on His ministry that is described in the first four books of the New Testament. His ministry culminated in His ultimate sacrifice, His own life, given to save humanity and unite us with God.
Christians are called to become more like Jesus each day. And sacrifice (giving something up) is a way to do this. We all have selfish tendencies. And in the Christian worldview these tendencies are the root of sin and divisiveness. So sacrifice counters those natural tendencies, roots out sin, and leads us on the path to spiritual growth.