Why don’t Catholics eat meat on Fridays during Lent, and why fish?

The tradition is to abstain from meat, not necessarily to eat fish.

Why Fast from Meat?

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.

The Law

Meat:

According to the USCCB.

“Abstinence laws consider that meat comes only from animals such as chickens, cows, sheep or pigs — all of which live on land. Birds are also considered meat“

And Fish:

Fish are a different category of animal. Salt and freshwater species of fish, amphibians, reptiles (cold-blooded animals) and shellfish are permitted.