Consider a commutative unital ring $R$ with a proper ideal $I.$ Consider the ideal $$J = \operatorname{rad}(I) = \{r \in R \,|\, r^n \in I \text{ for some integer } n \geq 1 \}.$$ Observe that in $R / J,$ the nilpotent elements are precisely the cosets whose representatives belong to $J.$ Explicitly, we have that $r + J$ is nilpotent if and only if $r^n + J = (r + J)^n = 0 + J$ for some integer $n \geq 1$ if and only if $r^n$ is in $J$ for some integer $n \geq 1$ if and only if $r \in J.$ Consequently, the nilradical of $R / J$ is precisely the set of elements in $J,$ i.e., the only nilpotent element of $R / J$ is the coset $0 + J.$ Consequently, the ring $R / J$ is reduced, i.e., there are no nonzero nilpotent elements in $R / J.$
On the other hand, the nilradical of a commutative unital ring $S$ is the intersection of all prime ideals in $S.$ Considering that every proper ideal (and hence every prime ideal) is contained in a minimal prime ideal, and every prime ideal contains a minimal prime ideal, it follows that the nilradical of $S$ is the intersection of all minimal prime ideals of $S,$ i.e., $\operatorname{nil}(S) = \bigcap_{P \in \operatorname{MinSpec}(S)} P,$ where we define$$\operatorname{MinSpec}(S) = \{P \,|\, P \text{ is a minimal prime ideal of } S \}.$$ By the previous paragraph, we have that $J = \bigcap_{P \in \operatorname{MinSpec}(R) \,|\, P \supseteq J} P = \bigcap_{P \in \operatorname{MinSpec}(R) \,|\, P \supseteq I} P.$