Hausdorf paracompact space is normal. Studied from Munkres' Topology its proof. He says that the Hausdorff condition enables us to choose, for each $b\in B$ an open set $U_b$ whose closure is disjoint from $a$. I did not get why disjoint closure is needed is it not enough to show that $U_b$ is disjoint from $a$? And similar question at the end when he says that closure of $V$ is disjoint from $a$. 

It may not be needed in some particular contexts, but it is exactly what the Hausdorff $T_2$ condition gives you: if you imagine an open neighbourhood $U_b$ disjoint from $a$ but whose closure contains $a$, then it is impossible to find a neighbourhood of $a$, $U_a$, such that $U_a\cap U_b=\emptyset$. 
The $T_2$ condition ensures that there should be such a $U_a$ for some $U_b$.
