The concepts in knowledge graphs (KGs) enable machines to understand natural language, and thus play an indispensable role in many applications. However, existing KGs have the poor coverage of concepts, especially fine-grained concepts. In order to supply existing KGs with more fine-grained and new concepts, we propose a novel concept extraction framework, namely MRC-CE, to extract large-scale multi-granular concepts from the descriptive texts of entities. Specifically, MRC-CE is built with a machine reading comprehension model based on BERT, which can extract more fine-grained concepts with a pointer network. Furthermore, a random forest and rule-based pruning are also adopted to enhance MRC-CE's precision and recall simultaneously. Our experiments evaluated upon multilingual KGs, i.e., English Probase and Chinese CN-DBpedia, justify MRC-CE's superiority over the state-of-the-art extraction models in KG completion. Particularly, after running MRC-CE for each entity in CN-DBpedia, more than 7,053,900 new concepts (instanceOf relations) are supplied into the KG. The code and datasets have been released at https://github.com/fcihraeipnusnacwh/MRC-CE