Objection 1. It would seem unfitting that those who fast should be bidden to abstain from flesh meat, eggs, and milk foods. For it has been stated above (Article 6) that fasting was instituted as a curb on the concupiscence of the flesh. Now concupiscence is kindled by drinking wine more than by eating flesh; according to Proverbs 20:1, "Wine is a luxurious thing," and Ephesians 5:18, "Be not drunk with wine, wherein is luxury." Since then those who fast are not forbidden to drink wine, it seems that they should not be forbidden to eat flesh meat.
Objection 2. Further, some fish are as delectable to eat as the flesh of certain animals. Now "concupiscence is desire of the delectable," as stated above (I-II, 30, 1). Therefore since fasting which was instituted in order to bridle concupiscence does not exclude the eating of fish, neither should it exclude the eating of flesh meat.
Objection 3. Further, on certain fasting days people make use of eggs and cheese. Therefore one can likewise make use of them during the Lenten fast.
On the contrary, stands the common custom of the faithful.
I answer that, As stated above (Article 6), fasting was instituted by the Church in order to bridle the concupiscences of the flesh, which regard pleasures of touch in connection with food and sex. Wherefore the Church forbade those who fast to partake of those foods which both afford most pleasure to the palate, and besides are a very great incentive to lust. Such are the flesh of animals that take their rest on the earth, and of those that breathe the air and their products, such as milk from those that walk on the earth, and eggs from birds. For, since such like animals are more like man in body, they afford greater pleasure as food, and greater nourishment to the human body, so that from their consumption there results a greater surplus available for seminal matter, which when abundant becomes a great incentive to lust. Hence the Church has bidden those who fast to abstain especially from these foods.
Reply to Objection 1. Three things concur in the act of procreation, namely, heat, spirit [Cf. P. I., Q. 118, 1, ad 3], and humor. Wine and other things that heat the body conduce especially to heat: flatulent foods seemingly cooperate in the production of the vital spirit: but it is chiefly the use of flesh meat which is most productive of nourishment, that conduces to the production of humor. Now the alteration occasioned by heat, and the increase in vital spirits are of short duration, whereas the substance of the humor remains a long time. Hence those who fast are forbidden the use of flesh meat rather than of wine or vegetables which are flatulent foods.
Reply to Objection 2. In the institution of fasting, the Church takes account of the more common occurrences. Now, generally speaking, eating flesh meat affords more pleasure than eating fish, although this is not always the case. Hence the Church forbade those who fast to eat flesh meat, rather than to eat fish.
Reply to Objection 3. Eggs and milk foods are forbidden to those who fast, for as much as they originate from animals that provide us with flesh: wherefore the prohibition of flesh meat takes precedence of the prohibition of eggs and milk foods. Again the Lenten fast is the most solemn of all, both because it is kept in imitation of Christ, and because it disposes us to celebrate devoutly the mysteries of our redemption. For this reason the eating of flesh meat is forbidden in every fast, while the Lenten fast lays a general prohibition even on eggs and milk foods. As to the use of the latter things in other fasts the custom varies among different people, and each person is bound to conform to that custom which is in vogue with those among whom he is dwelling. Hence Jerome says [Augustine, De Lib. Arb. iii, 18; cf. De Nat. et Grat. lxvii]: "Let each province keep to its own practice, and look upon the commands of the elders as though they were the laws of the apostles."
Source: Summa Theologica (New Advent)