The 'eye for an eye' principle puts to right the wrong that a person did.
The wrongdoer is punished proportionately for what he or she did wrong.
There must be no vengeance in that biblical principle.
There is nothing wrong with giving proportionate punishment to a law-breaker.
But vengeance is based on spite and hatred and a desire to not just 'get even' with the one you are offended against, but to go one better (or worse, actually). It is wrong, in itself, and creates more harm, more ill-will, and on-going desire to perpetuate the retaliation. That is how feuds get going, and may last for centuries. If you cannot see how wrong that is, then you have no thought for anything but to satisfy your injured pride.