With the integration of high shares of distributed generators (DGs), it is increasingly difficult to cope with the various uncertainties and puts forward higher requirements for the operational flexibility in active distribution networks (ADNs). Controllability of ADNs has been significantly improved by the application of energy storage and power electronic devices. However, due to the difficulties in effective coordination of various controllable resources, the controllability cannot be fully translated into the system flexibility. In this paper, an analytical framework for quantifying the flexibility of ADNs is proposed, including the quantification of node flexibility, the matching of system flexibility, and the flexibility of network transmission. The proposed framework provides a novel perspective of flexibility to reinterpret operation issues of distribution networks. As the power imbalance and voltage deviation are even more severe caused by the high shares of DG integration, the indexes of flexibility from magnitude, frequency and intensity dimensions are deteriorated in ADNs operation. Through the spatial and temporal regulation of power flow, various controllable devices, such as soft open points (SOPs) and energy storage system (ESS), can effectively mitigate the power imbalance and voltage deviation to improve the indexes of system flexibility. Thus, under the unified analytical framework, the potential benefits of system controllability are fully utilized to provide effective countermeasures for the flexible operation of ADNs. Finally, case studies are performed on the modified IEEE 33-node system to quantify the operational flexibility of ADNs and verify the flexibility enhancement brought by controllable resources.