Let $A\overset{f}{\to} B$ be a continuous map with $B$ sufficiently connected to admit a universal cover $\widetilde B$. Consider the covering map $\widetilde B/f_\ast\pi_1(A,a)\to B$. The fundamental group of the quotient is $f_\ast\pi_1(A,a)$, so the lifting criterion ensures $f$ factors as $A\to\widetilde B/f_\ast\pi_1(A,a)\to B$.
Why is the first factor of this factorization a 1-connected map (equivalently, why does it have connected homotopy fibers)?
I think $A$ should be path connected if we wish to use the lifting criterion.
So $\bar{f}$, the lift, induces a bijection on $\pi_0$ because the universal cover is path connected, hence this quotient is. On $\pi_1$, the map is surjective since the covering map induces the inclusion of $f_*\pi_1(A)$ and obviously $\pi_1(A)$ surjects onto $f_*\pi_1(A)$