Mark 2:18–22
18 Now John’s disciples and the Pharisees were fasting. And people came and said to him, “Why do John’s disciples and the disciples of the Pharisees fast, but your disciples do not fast?” 19 And Jesus said to them, “Can the wedding guests fast while the bridegroom is with them? As long as they have the bridegroom with them, they cannot fast. 20 The days will come when the bridegroom is taken away from them, and then they will fast in that day. 21 No one sews a piece of unshrunk cloth on an old garment. If he does, the patch tears away from it, the new from the old, and a worse tear is made. 22 And no one puts new wine into old wineskins. If he does, the wine will burst the skins—and the wine is destroyed, and so are the skins. But new wine is for fresh wineskins.”
Fasting was a regular religious practice in Judaism. The three main pillars of Judaism were prayer, fasting and alms giving. The people were only required to fast one day a year—the Day of Atonement, yet they often fasted at other times throughout their history. The Mishnah (a book of Jewish religious practices) prescribed fasts during times of lament over national tragedies, times of crisis, as well as personal fasts for a number of reasons. Vows were confirmed by fasting, remorse and penitence were accompanied with fasting, and prayer was supported by fasting. In Jesus' day the Pharisees, and apparently John the Baptist's disciples fasted often. The Pharisees would fast ever Monday and Thursday from dawn until dusk. Although this wasn't a requirement it was often expected as a sign of religious piety.
Note here Jesus doesn't abolish the idea of fasting. Instead he suggests that his followers will fast, just not while he is with them. Now is a time for action, not remorse and lament. In Matthew 6:16 Jesus says, "When you fast...", implying that his followers will fast. Followers of Jesus, however, are not to fast as an external sign of piety like the Pharisees, instead they are to fast out of a genuine heart towards God.
Fasting ought to be a regular practice in the Christian life out of a pure heart to commune with God. Fasting teaches us to discipline our body. If we can abstain from food, we can abstain from sin. It creates time, time that we would've otherwise spent prepping food, for prayer. It experientially teaches us that, "Man does not live on bread alone, but by every word that comes from the mouth of God." (Matt. 4:4) It humbles us to remember our dependence on God's provision.