Gravity may be ``locally localized'' over a wide range of length scales on a d-dimensional anti-de Sitter (AdS) brane living inside AdS<sub>d+1</sub>. In this paper we examine this phenomenon from the point of view of the holographic dual ``defect conformal field theory''. The mode expansion of bulk fields on the gravity side is shown to be precisely dual to the ``boundary operator product expansion'' of operators as they approach the defect. From the field theory point of view, the condition for localization is that a ``reduced operator'' appearing in this expansion acquires negative anomalous dimension. In particular, a very light localized graviton exists when a mode arising from the reduction of the ambient stress-energy tensor to the defect has conformal dimension Δ ~ d−1. The part of the stress tensor containing the defect dynamics has dimension Δ = d−1 in the free theory, but we argue that it acquires a positive anomalous dimension in the interacting theory, and does not participate in localization in the regime of small backreaction of the brane. We demonstrate that such an anomalous dimension is consistent with the conservation of the full stress-energy tensor. Finally, we analyze how to compute the anomalous dimensions of reduced operators from gravity at leading order in the interactions with the brane.