Why is parameter $q$ of $U_q(mathfrak{sl}_2)$ being root of unity a problem?

I am recently studying the quantum groups and the exactly solvable models.

I understand the solution of the Yang-Baxter equation is elements in $U_q(mathfrak{sl}_2)$.

I am informed that when $q$ is the root of unity, there is a massive center in this quantum group.

However, I fail to see that.

And I also very much confused that what is the terrible consequence of this massive center?
Or what is the meaning of center of quantum groups in physics?