I'm just wanting to clarify the rule about eating fish on Good Friday: Is the intention to add fish to the diet or to remove all other meats from the diet?
is the intention to add fish to the diet or to remove all other meats from the diet?
The answer is "Neither". The intention is to remind you of your need for, and dependence on God.
Abstaining from meat is a specific example of the teaching on Abstinence. The Catholic Encyclopedia explains the purpose of abstinence in general as follows:
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 (Genesis 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. The hour which witnessed the transgression of this law marked an increase in the debt which the creature owed the Creator. Adam's disobedience rendered all men criminal, and liable to the necessity of appeasing God's justice. To meet this new exigency nature dictated the necessity of penance; positive legislation determined the ways and means whereby this natural obligation would best be concreted. The chief results of this determination are positive statutes concerning fasting and abstinence. Laws relating to fasting are principally intended to define what pertains to the quantity of food allowed on days of fasting, while those regulating abstinence, what refers to the quality of viands. In some instances both obligations coincide; thus, the Fridays of Lent are days of fasting and abstinence. In other instances the law of abstinence alone binds the faithful; thus ordinary Fridays are simply days of abstinence. The purpose of this article is to trace the history of ecclesiastical legislation regarding the law of abstinence, as well as to examine the motives which underlie this legislation.
Short version: Abstinence is a form of penance, the purpose of which is to remind us of our dependence on our Creator, God.
If your focus is "why meat, specifically", that is answered, as @Andrewleach pointed out, at svidgen's answer this question:
Short version, meat was exceptional, and the idea was to abstain from something that we enjoy the most.
I'm not going to steal his quotes or answer, but if that's the angle your most concerned with, vote that answer up.