Let  be a finite simple Lie algebra, and let r denote the ratio of the square length of long roots to that of short roots. Let p>2r be an integer and  a primitive p-th root of unity. Denote by  the Lusztig big quantum affine algebra at root of unity defined by divided powers. In this paper, we establish a current algebra presentation of  . Based on this presentation, we construct a - module quantum vertex algebras  for each integer . Moreover, we establish a fully faithful functor from the category of smooth weighted -modules of level  to the category of -equivariant -coordinated quasi-modules of , where  is the group homomorphism defined by . We also determine the image of this functor. The structure  is substantially different from that of affine vertex algebras. We realize  as a deformation of a simpler quantum vertex algebra  by using vertex bialgebras, and decompose  into a Heisenberg vertex algebra and a more interesting quantum vertex algebra determined by a quiver.