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?
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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:
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.
The Catholic Church's law prohibits eating meat on days of complete abstinence. Fish, and other seafood, are permitted and are therefore often eaten on those days, but they are not in any way required. An entirely vegetarian or even vegan diet would comply with the law of abstinence.