In the secular world, ‘fasting’ means abstaining from all food and drink, but in religious circles, ‘fasting’ means going on a disciplined diet. The purpose of a fast is to find out who is in control, you or your belly, and to win that control if necessary. It's also a way of using your appetite as a spiritual snooze alarm that moves you to pray.
So now that we understand that fasting means a diet and not total deprivation, I can answer the question.
In the first century, Jews fasted on Mondays and Thursdays. The original Christians were all Jewish and were used to the fasting as a spiritual discipline. They moved the fast days to Wednesdays and Fridays, because Judas engineered Jesus' arrest on a Wednesday and Jesus was crucified on a Friday. Most often that fast took the form of avoiding meat in the diet. In those days, meat was a luxury food. You either had to buy it in a market or you had to own enough land to keep cattle. On the other hand, anyone could grow vegetables or forage for them, and anyone could catch a fish in a lake or a stream. You could buy better fish and vegetables, but the point is that you could eat without money if you were poor. So meat was rich people's food and fish was poor people's food. That is why the most common form of fasting was to omit meat and eat fish.
The Wednesday and Friday fasts were a universal Christian custom in ancient times. The Eastern Orthodox still observe these fasts. The Roman Catholic Church downplayed the Wednesday fast, but kept the Friday fast until quite recently. Anglicans and Protestants also observed these fasts. In the 18th century, a man could not be ordained a Methodist minister if he did not fast on Wednesdays and Fridays, with the reasoning that a person who could not rule his own belly could certainly not rule the church.
I have been able to trace back the Season of Lent to at least the third century. During Lent, people fasted on every day except Sunday in the West and every day but Saturday and Sunday in the East. Older cookbooks have special recipes for Lent, and you can still buy Lenten cookbooks from Eastern Orthodox publishers, such as Light and Life.
The fasts have died out in the west for several reasons. First, we are becoming very narcissistic. We don't care if our bodily appetites have the upper hand, and many people count recreation and luxury as necessities. In the Roman Catholic Church in the United States, the bishops relaxed the fasts because they were not being observed and also because they realized that Lobster Thermidor or even a modest dinner at a seafood restaurant hardly lives up to the spirit of a fast. I say the bishops relaxed the fasts, because in the Roman Catholic Church, the regulations for fasting are set by the national conferences of bishops and can vary from place to place.
Nothing in the Bible absolutely requires us to fast. However, when Jesus discusses fasting in Matthew 6, He clearly assumes we have a discipline of fasting as part of our spirituality. He does not say, “If you should happen to choose the option of fasting,” He says, “When you fast”:
When you fast, do not look somber as the hypocrites do, for they disfigure their faces to show men they are fasting. I tell you the truth, they have received their reward in full. But when you fast, put oil on your head and wash your face, so that it will not be obvious to men that you are fasting, but only to your Father, who is unseen; and your Father, who sees what is done in secret, will reward you.
—Matthew 6:16-18, NIV
In another place, Jesus said that while He was present in the flesh, His disciples did not observe the fasts (meaning probably the Monday and Thursday fasts), but that we would resume fasting after He ascended.
Then John's disciples came and asked him, “How is it that we and the Pharisees fast, but your disciples do not fast?”
Jesus answered, “How can the guests of the bridegroom mourn while he is with them? The time will come when the bridegroom will be taken from them; then they will fast.”
—Matthew 9:14-15, NIV
Historically, Christians did resume fasting after Jesus’ Ascension, moving the fasts to Wednesday (the day of the betrayal) and Friday (the day of the crucifixion) and we remained faithful up to the present day.