The conifold is the space of vacua of the Klebanov-Witten theory. It is obtained as the quotient of the complex four dimensional space of fields $A_1$,$A_2$,$B_1$,$B_2$ by the action of the gauge group $U(1) \times U(1)$. In this action, the diagonal $U(1)$ fixes everyone and only the point $(0,0,0,0)$ is fixed by the full $U(1) \times U(1)$. This implies that the gauge group of the theory is $U(1) \times U(1)$ at the singular point of the conifold and is Higgsed to the diagonal $U(1)$ for a point away from the singularity. At the singular point, the four fields $A_1$,$A_2$,$B_1$, $B_2$ can freely fluctuate despite the fact that the conifold is complex three dimensional: it is the definition of a singular point to have a tangent space of dimension, here complex four, greater than the dimension of the manifold. At a point away from the singularity, the four fields $A_1$,$A_2$,$B_1$, $B_2$ are in the adjoint of the diagonal $U(1)$ (of course, adjoint of $U(1)$ is trivial but I keep a terminology which works for the general $U(N)$ case) but one linear combination is frozen because the tangent space at such a point is complex three dimensional (for example, if $A_1 B_1$ is the only combination with a non-trivial expectation value, computation of the tangent space shows that the $A_2 B_2$ direction is frozen). This implies that the theory at this point is a $U(1)$ gauge theory with three massless chiral multiplets in the adjoint representation. This is exactly the content of a $U(1)$ $N=4$ vector multiplet (indeed, a $N=4$ vector multiplet is the same thing as a $N=2$ vector multiplet with a massless $N=2$ hypermultiplet in the adjoint, which is the same thing as a $N=1$ vector multiplet with three massless $N=1$ chiral multiplets in the adjoint).
The $U(N)$ case is similar. The only new thing is to check that the superpotential of the Klebanov-Witten theory reduces to the superpotential of the $N=4$ super Yang-Mills at a Higgsed point of the moduli space.