The requirements for fasting have changed over the years in order to accomodate various times and cultures. Currently Catholics in America are required to fast the whole day on Ash Wednesday and Good Friday. This means we are to eat only one meal, but we can also have two snacks during the day that don't add up to one meal. These are also days of abstinence, requiring us to refrain from eating meat. We are also required to fast one hour before mass, meaning only water and any medicine needed is allowed. Abstinence from meat is required during the Fridays of Lent, and some kind of sacrifice is encouraged for Fridays throughout the whole year. (As many know, it used to be that abstinence was required on all Fridays of the year. The Church may change such disciplines the way a parent may change a curfew for their child if they feel it proper.) Of course there are exceptions to the fasting requirements such as the young, elderly, sick, and pregnant.
The Catechism explains that fasting helps us to "acquire mastery over our instincts and freedom of heart" (CCC 2043). One of the most frequent messages of Christ is that in order to be truly free, we must die to self. The idea of "freedom" as being able to do whatever we want is an illusion, because "doing whatever we want" typically means giving into our cravings, desires, and passions without restraint. In doing so, we actually become slaves to our desires, unwilling and for some unable to say no to them.
I like how simply and yet firmly the Baltimore Catechism puts it:
Q. 1346. Why does the Church command us to fast and abstain?
A. The Church commands us to fast and abstain, in order that we may mortify our passions and satisfy for our sins.
Q. 1347. What is meant by our passions and what by mortifying them?
A. By our passions are meant our sinful desires and inclinations. Mortifying them means restraining them and overcoming them so that they have less power to lead us into sin.
The scriptures say that after Jesus fasted in the desert for 40 days, he was hungry. This seems kind of obvious, but I think it actually makes an important point. Hunger is an emptiness, it's a readiness to be filled. And this is why we fast - to empty ourselves so that we may be filled with Christ. We must decrease so that Christ may increase.
Then Jesus was led by the Spirit into the desert to be tempted by the devil. He fasted for forty days and forty nights, and afterwards he was hungry. (Matt 4:1-2)
When you fast, do not look gloomy like the hypocrites. They neglect their appearance, so that they may appear to others to be fasting. Amen, I say to you, they have received their reward. But when you fast, anoint your head and wash your face, so that you may not appear to be fasting, except to your Father who is hidden. And your Father who sees what is hidden will repay you. (Matt 6:16-18)
Then the disciples of John approached him and said, "Why do we and the Pharisees fast (much), but your disciples do not fast?" Jesus answered them, "Can the wedding guests mourn as long as the bridegroom is with them? The days will come when the bridegroom is taken away from them, and then they will fast. (Matt 9:15)
Anna, the daughter of Phanuel, of the tribe of Asher...never left the temple, but worshiped night and day with fasting and prayer. (Luke 2:36-7)
While they were worshiping the Lord and fasting, the holy Spirit said, "Set apart for me Barnabas and Saul for the work to which I have called them." Then, completing their fasting and prayer, they laid hands on them and sent them off. (Acts 13:2-3)
They appointed presbyters for them in each church and, with prayer and fasting, commended them to the Lord in whom they had put their faith. (Acts 14:23)