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 A1,A2,B1,B2 by the action of the gauge group U(1)×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)×U(1). This implies that the gauge group of the theory is U(1)×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 A1,A2,B1, B2 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 A1,A2,B1, B2 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 A1B1 is the only combination with a non-trivial expectation value, computation of the tangent space shows that the A2B2 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.