Fog radio access networks (F-RANs), which consist of a cloud and multiple edge nodes (ENs) connected via fronthaul links, have been regarded as promising network architectures. The F-RAN entails a joint optimization of cloud and edge computing as well as fronthaul interactions, which is challenging for traditional optimization techniques. This paper proposes a Cloud-Enabled Cooperation-Inspired Learning (CECIL) framework, a structural deep learning mechanism for handling a generic F-RAN optimization problem. The proposed solution mimics cloud-aided cooperative optimization policies by including centralized computing at the cloud, distributed decision at the ENs, and their uplink-downlink fronthaul interactions. A group of deep neural networks (DNNs) are employed for characterizing computations of the cloud and ENs. The forwardpass of the DNNs is carefully designed such that the impacts of the practical fronthaul links, such as channel noise and signling overheads, can be included in a training step. As a result, operations of the cloud and ENs can be jointly trained in an end-to-end manner, whereas their real-time inferences are carried out in a decentralized manner by means of the fronthaul coordination. To facilitate fronthaul cooperation among multiple ENs, the optimal fronthaul multiple access schemes are designed. Training algorithms robust to practical fronthaul impairments are also presented. Numerical results validate the effectiveness of the proposed approaches.