The distributed renewable energy generations, as accessible and easily targets for attackers, introduce an extra false data injection attack (FDIA) threat in the smart distribution networks. Scattered attack points and complex attack features hinder the elimination of potential threats. In this context, an FDIA fast detection and pinpoint localization framework is proposed. This framework identifies abnormal signals and attacked nodes from the unique topology structure and status contiguity of smart distribution networks, namely, spatial-temporal correlations of power grids, by using a cluster partition-fuzzy broad learning system (CP-FBLS). Unlike most existing FDIA detection methods, which are dedicated to high accuracy but neglect the urgent need for rapid detection in smart distribution networks, the proposed CP-FBLS framework maintains the fast computational nature of a fuzzy broad learning system (FBLS), while avoiding the accuracy degradation caused by high-dimension of data in large-scale smart distribution networks. Moreover, the multi-layer structure of the proposed framework recognizes the location of FDIA, bridging the research gap of attack localization. To comprehensively evaluate the proposed strategy, datasets containing various FDIA types are constructed. Numerical simulations based on the above datasets in modified IEEE 34-bus and 123-bus distribution systems are implemented. The results of the case studies showed that the proposed method can achieve 98.43% accuracy with 0.34ms detection time, realizing rapid detection and localization of various FDIAs with satisfactory accuracy.