Fasting is something that Jesus did, and that he expected of his followers - 'when you fast' (Matthew 6:16). It was something which was part of normal spiritual life in Jesus day, and though he questioned the way in which some people fasted, he never queried the practice itself. It's therefore part of the normal Christian life.
There seem to be a cluster of different reasons for fasting in the Scriptures:
1. Obedience - Jesus expects it, so we do it. There are some things it's good to just do because we're being obedient, rather than because we expect any benefit. Self-centredness needs to be fasted from too!
2. Intensity in prayer: Nehemiah fasts and prays before going to see the king about rebuilding Jerusalem. The Ninevites fast when repenting at Jonah's preaching. Fasting adds sincerity and weight to prayer, it's both a sign (to God and to ourselves) that we're serious about it, and it can make prayer itself more powerful and intense.
3. Spiritual battle: Jesus major fast - the 40 days in the wilderness - brings him into close quarters with the devil and his temptations. Sometimes fasting is an intense assault on evil, either within ourselves or within society.
4. Training: fasting trains our body to do without food, by exercising willpower and self-denial, we increase the strength our will, and our capacity to deny ourselves. It is a spiritual discipline, which makes us spiritually fitter
5. Physical well being: Daniel and his friends abstained from meat and rich food, and were physically better for it. This isn't a prime reason for fasting, but during the fasting process (especially longer fasts) toxins and other nasty stuff in the body is broken down and got rid of. In the short run, fasting from addictive substances like caffeine will give us a headache, but will help to break physical dependence.
Note: There are various medical reasons not to fast, so make sure you check these out before you think about it, and it's sensible to take some tips either from the books, or other people who've done it, before you start. Health and Safety announcement over!
If you're serious about fasting, then the best place to start is Richard Fosters classic 'Celebration of Discipline' chapter 4. There are also various online resources, (e.g. here, here, and an article from the Renovare website which puts spiritual disciplines in a wider context). Start small, drink lots of water, try to use the time freed up from preparing food/eating to pray, and be aware when you're with others that you might get stinky breath!