Why should we fast?

We fast for many reasons: to imitate Christ, for penance, and to conquer ourselves among them. And we fast out of obedience: Our Lord and His Apostles tell us to. This last reason is described well by Pope Clement XIII in his "Appetente Sacro," written in 1759. In this document, he exhorts his Bishops to explain to their flocks the reasons for fasting:

You will begin most appropriately, and with hope of the greatest profit, to recall men to the observance of the holy law of fasting, if you teach the people this: penance for the Christian man is not satisfied by withdrawing from sin, by detesting a past life badly lived, or by the sacramental confession of these same sins. Rather, penance also demands that we satisfy divine justice with fasting, almsgiving, prayer, and other works of the spiritual life. Every wrongdoing -- be it large or small -- is fittingly punished, either by the penitent or by a vengeful God. Therefore we cannot avoid God's punishment in any other way than by punishing ourselves. If this teaching is constantly implanted in the minds of the faithful, and if they drink deeply of it, there will be very little cause to fear that those who have discarded their degraded habits and washed their sins clean through sacramental confession would not want to expiate the same sins through fasting, to eliminate the concupiscence of the flesh. Besides, consider the man who is convinced that he repents of his sins more firmly when he toes not allow himself to go unpunished. That man, already consumed with the love of penance, will rejoice during the season of Lent and on certain other days, when the Church declares that the faithful should fast and gives them the opportunity to bring forth worthy fruits of penance.

 
The Proper attitude when fasting

St. John Chrysostom, in this excerpt from Homily III of his "Homilies on the Statues," summed it up well:

7. ...We have this fast too as an ally, and as an assistant in this good intercession. Therefore, as when the winter is over and the summer is appearing, the sailor draws his vessel to the deep; and the soldier burnishes his arms, and makes ready his steed for the battle; and the husbandman sharpens his sickle; and the traveller boldly undertakes a long journey, and the wrestler strips and bares himself for the contest. So too, when the fast makes its appearance, like a kind of spiritual summer, let us as soldiers burnish our weapons; and as husbandmen let us sharpen our sickle; and as sailors let us order our thoughts against the waves of extravagant desires; and as travellers let us set out on the journey towards heaven; and as wrestlers let us strip for the contest. For the believer is at once a husbandman, and a sailor, and a soldier, a wrestler, and a traveller. Hence St. Paul saith, "We wrestle not against flesh and blood, but against principalities, against powers. Put on therefore the whole armour of God." Hast thou observed the wrestler? Hast thou observed the soldier? If thou art a wrestler, it is necessary for thee to engage in the conflict naked. If a soldier, it behoves thee to stand in the battle line armed at all points. How then are both these things possible, to be naked, and yet not naked; to be clothed, and yet not clothed! How? I will tell thee. Divest thyself of worldly business, and thou hast become a wrestler. Put on the spiritual armour, and thou hast become a soldier. Strip thyself of worldly cares, for the season is one of wrestling. Clothe thyself with the spiritual armour, for we have a heavy warfare to wage with demons. Therefore also it is needful we should be naked, so as to offer nothing that the devil may take hold of, while he is wrestling with us; and to be fully armed at all points, so as on no side to receive a deadly blow. Cultivate thy soul. Cut away the thorns. Sow the word of godliness. Propagate and nurse with much care the fair plants of divine wisdom, and thou hast become a husbandman. And Paul will say to thee, "The husbandman that laboureth must be first partaker of the fruits. He too himself practised this art. Therefore writing to the Corinthians, he said, "I have planted, Apollos watered, but God gave the increase." Sharpen thy sickle, which thou hast blunted through gluttony--sharpen it by fasting. Lay hold of the pathway which leads towards heaven; rugged and narrow as it is, lay hold of it, and journey on. And how mayest thou be able to do these things? By subduing thy body, and bringing it into subjection. For when the way grows narrow, the corpulence that comes of gluttony is a great hindrance. Keep down the waves of inordinate desires. Repel the tempest of evil thoughts. Preserve the bark; display much skill, and thou hast become a pilot. But we shall have the fast for a groundwork and instructor in all these things.

8. I speak not, indeed, of such a fast as most persons keep, but of real fasting; not merely an abstinence from meats; but from sins too. For the nature of a fast is such, that it does not suffice to deliver those who practice it, unless it be done according to a suitable law. "For the wrestler," it is said, "is not crowned unless he strive lawfully." To the end then, that when we have gone through the labour of fasting, we forfeit not the crown of fasting, we should understand how, and after what manner, it is necessary to conduct this business; since that Pharisee also fasted, but afterwards when down empty, and destitute of the fruit of fasting. The Publican fasted not; and yet he was accepted in preference to him who had fasted; in order that thou mayest learn that fasting is unprofitable, except all other duties follow with it. The Ninevites fasted, and won the favour of God. The Jews fasted too, and profited nothing, nay they departed with blame. Since then the danger in fasting is so great to those who do not know how they ought to fast, we should learn the laws of this exercise, in order that we may not "run uncertainly," nor "beat the air," nor while we are fighting contend with a shadow. Fasting is a medicine; but a medicine, though it be never so profitable, becomes frequently useless owing to the unskillfulness of him who employs it. For it is necessary to know, moreover, the time when it should be applied, and the requisite quantity of it; and the temperament of body that admits it; and the nature of the country, and the season of the year; and the corresponding diet; as well as varous other particulars; any of which, if one overlooks, he will mar all the rest that have been named. Now if, when the body needs healing, such exactness is required on our part, much more ought we, when our care is about the soul, and we seek to heal the distempers of the mind, to look, and to search into every particular with the utmost accuracy.

11. I have said these things, not that we may disparage fasting, but that we may honour fasting; for the honour of fasting consists not in abstinence from food, but in withdrawing from sinful practices; since he who limits his fasting only to an abstinence from meats, is one who especially disparages it. Dost thou fast? Give me proof of it by thy works! Is it said by what kind of works? If thou seest a poor man, take pity on him! If thou seest an enemy, be reconciled to him! If thou seest a friend gaining honour, envy him not! If thou seest a handsome woman, pass her by! For let not the mouth only fast, but also the eye, and ear, and the feet, and the hands, and all the members of our bodies. Let the hands fast, by being pure from rapine and avarice. Let the feet fast, but ceasing from running to the unlawful spectacles. Let the eyes fast, being taught never to fix themselves rudely upon handsome countenances, or to busy themselves with strange beauties. For looking is the food of the eyes, but if this be such as is unlawful or forbidden, it mars the fast; and upsets the whole safety of the soul; but if it be lawful and safe, it adorns fasting. For it would be among things the most absurd to abstain from lawful food because of the fast, but with the eyes to touch even what is forbidden. Dost thou not eat flesh? Feed not upon lasciviousness by means of the eyes. Let the ear fast also. The fasting of the ear consists in refusing to receive evil speakings and calumnies. "Thou shalt not receive a false report," it says.

 
Footnote:
¹ During Embertides, the traditional practice is to revert to the older pattern of abstaining and fasting on Wednesdays in addition to Fridays. Saturdays are added, too, during these times.

² Fridays are often called "Fish Fridays" not because Catholics must eat fish on those days, but because it's permissible to eat fish, and eating fish has become a small-T tradition. It's because of the Catholic practice of abstaining from meat that McDonalds restaurants sell fish fillet sandwiches. From a Standpoint magazine article, "Dechristianity by the numbers," written by Damian Thompson:

In 1959, Lou Groen, president of the Cincinnati Restaurant Association, opened the first branch of McDonald’s in the area. Immediately he ran into a problem: sales of hamburgers dropped sharply every Friday. That was because the restaurant was in Montfort Heights, a suburb full of Catholics. Sixty years ago, no Mass-goer would have dreamt of eating meat on Fridays. So Groen, himself a practising Catholic, approached Ray Kroc, the control-freak founder of the McDonald’s franchise, who flew into a rage if unused ketchup sachets were thrown away and spent his last years spying on his local McDonald’s with a telescope.

Groen suggested that his branch should sell a non-meat sandwich, with fish instead of beef in the middle. That was a brave thing to do. Old Ray had an almost religious devotion to McDonald’s hamburgers, finding “grace in the texture and softly curved silhouette of a bun”. He didn’t want “my stores stunk up with the smell of fish”, but he agreed to an experiment. On Good Friday, 1962, Groen sold his fish sandwich alongside an alternative of Kroc’s own devising: the Hula Burger, which substituted a slice of pineapple for the beef. The Catholics of Montfort Heights chose fish over pineapple by an enormous margin. And so the Filet-O-Fish was born, and survives today—unlike the Hula Burger, which Kroc launched nationally and then had to kill off quietly.