Inductive knowledge graph completion (KGC) aims to predict missing triples with unseen entities. Recent works focus on modeling reasoning paths between the head and tail entity as direct supporting evidence. However, these methods depend heavily on the existence and quality of reasoning paths, which limits their general applicability in different scenarios. In addition, we observe that latent type constraints and neighboring facts inherent in KGs are also vital in inferring missing triples. To effectively utilize all useful information in KGs, we introduce CATS, a novel context-aware inductive KGC solution. With sufficient guidance from proper prompts and supervised fine-tuning, CATS activates the strong semantic understanding and reasoning capabilities of large language models to assess the existence of query triples, which consist of two modules. First, the type-aware reasoning module evaluates whether the candidate entity matches the latent entity type as required by the query relation. Then, the subgraph reasoning module selects relevant reasoning paths and neighboring facts, and evaluates their correlation to the query triple. Experiment results on three widely used datasets demonstrate that CATS significantly outperforms state-of-the-art methods in 16 out of 18 transductive, inductive, and few-shot settings with an average absolute MRR improvement of 7.2%.