Goel Hadam

I am currently teaching Sefer Dvarim to high school seniors. We are discussing the mitzvah of Ir Miklat, and with it the concept of a Goel Hadam, a person who is legitimately allowed to pursue and kill the murderer, seemingly in order to enact revenge. any thoughts as to why the Torah allows this in this situation while in other places in the Torah we are taught not to take revenge?

Thanks so much,
Sima Neiger