nicbou's link has details on the how, but here's a direct answer to the why:
The money is not a tax because it's not the government that collects it, and the government cannot decide how much it should be. Instead, there are institutions that have been granted powers to levy their own fees and self-govern to a large degree. Accountability is established through an oversight board that includes representatives from civic society and other independent review institutions.
Worth adding that it's a compulsory charge for every German household, regardless of whether they watch or listen to the programming, or own a TV. Non-payment leads to collection agency, and ultimately jail.
That accountability is more of a joke though.
It's a very self-serving system and the people on top are granting themselves more lavish salaries than what the chancellor gets.
The money is not a tax because it's not the government that collects it, and the government cannot decide how much it should be. Instead, there are institutions that have been granted powers to levy their own fees and self-govern to a large degree. Accountability is established through an oversight board that includes representatives from civic society and other independent review institutions.