Binary Code Similarity Detection, essential for binary-code related tasks like vulnerability detection, has attracted increasing attention in recent years. However, existing methods frequently fall short of achieving both high precision and recall at scale, and their results often lack interpretability due to the neglect of function context and reliance on purely similarity-driven outputs. Our key insights are twofold: \emph{1) Binary functions are not self-contained; they depend on other code and data beyond their content to fulfill their functionalities. 2) Large language models (LLMs) excel not only at analyzing code but also at generating reasonable explanations.} Motivated by these insights, we propose a general BCSD framework, Co$^2$FuLL. We first systematically select stable and representative code and data features, along with their corresponding dependencies on the functions, to construct the function context. Then, by fusing function context with content similarities computed by the existing BCSD approach, we substantially narrow down the search space. Ultimately, we employ LLMs with a carefully designed prompt to verify the remaining candidates and produce clear, human-readable explanations. We conduct comprehensive experiments on a large function pool under varying compilation settings and after binary stripping. The results show that Co$^2$FuLL based on HermesSim and DeepSeek-V3 achieves 80.5% precision and 94.4% recall, improving the baseline HermesSim by 142.5% and 42.2%, respectively, providing an accurate and interpretable solution for BCSD.