Why existence of universal covering implies that the base space be locally path connected?

It depends what the definition of "universal covering" is.

If the definition is that a universal covering of $B$ is a simply connected covering (as it is in Munkres), then this statement as you're writing it isn't quite true. If $B$ is path-connected and simply connected but not locally path connected, then the identity function $B\to B$ is the universal covering of $B$ (but it exists!). For example, let B be the union of the line segments in $\mathbb{R}^2$ from $(0,0)$ to each point of $\{(1,0),...,(1,1/3),(1,1/2),(1,1)\}$.

But if I recall correctly, Munkres typically assumes spaces in consideration are locally path connected. So what is meant is the following: A path-connected and locally path-connected space $B$ has a universal covering if and only if $B$ is semilocally simply connected.