Why don’t Catholics eat meat on Fridays during Lent?
The Catholic Encyclopedia states:
Inasmuch as abstinence signifies abstaining from food, the Bible narrative points to the first instance wherein such a course of conduct was imposed by law (Gen. 2:16–17). The obvious purpose of this mandate was to lead the moral head of the human race to recognize the necessary dependence of creature upon Creator. . . . Throughout the Latin church the law of abstinence prohibits all responsible subjects from indulging in meat diet on duly appointed days. Meat diet comprises the flesh, blood, or marrow of such animals and birds as constitute flesh meat according to the appreciation of intelligent and law-abiding Christians.
In the ancient world, meat was often considered to be an indulgence, so abstaining specifically from meat on days of fast and abstinence was intended as a way for Christians to deny themselves an indulgence in recognition of the supreme sacrifice offered by Christ on Good Friday. As to why fish ordinarily is allowed on days of fast and abstinence, I recommend reading Breakfast with the Lord, a blog post I wrote on the topic.