According to the edit, the essential question is why the Church prescribes a specific penitential action (abstaining from meat) on specific days, rather than just telling us to do penance (our own way). I think the reason for being so specific is that, although a general exhortation to do penance would be followed by some people, it would fail for a great many other people, even well-meaning people who intend to do penance. Penance is too easily procrastinated. If I eat meat (knowingly) on a day of abstinence, I know that I've committed a sin that I need to confess. If I procrastinate a general intention to do penance, it never quite feels like a real sin --- after all, I'm planning to do some penance next week or next month or whenever.
The preceding paragraph was about why the Church makes specific rules. Another aspect of the question is why the rule refers specifically to abstaining from meat. I don't know the historical reason for that, but here's a guess. For many centuries, meat was not part of the regular diet of "ordinary" folks; only the rich could afford to eat meat often. So the requirement of abstinence had little or no effect for most people, but it required the rich to occasionally live like the rest of the population, an exercise in humility.
Does this guess vitiate my first paragraph? After all, ordinary people also need to do penance, yet, if I'm right, they weren't seriously affected by the law of abstinence. The answer to that is that abstinence from meat was not the only penitential practice prescribed by the Church. There are also days (and a whole season --- Lent) of fasting, which imposes a limit on the amount of food that one is allowed to eat. That would affect everyone, not just the rich. (And the restrictions were more severe longer ago).