Efficient collaboration between collaborative machine learning and wireless communication technology, forming a Federated Edge Learning (FEEL), has spawned a series of next-generation intelligent applications. However, due to the openness of network connections, the FEEL framework generally involves hundreds of remote devices (or clients), resulting in expensive communication costs, which is not friendly to resource-constrained FEEL. To address this issue, we propose a distributed approximate Newton-type algorithm with fast convergence speed to alleviate the problem of FEEL resource (in terms of communication resources) constraints. Specifically, the proposed algorithm is improved based on distributed L-BFGS algorithm and allows each client to approximate the high-cost Hessian matrix by computing the low-cost Fisher matrix in a distributed manner to find a "better" descent direction, thereby speeding up convergence. Second, we prove that the proposed algorithm has linear convergence in strongly convex and non-convex cases and analyze its computational and communication complexity. Similarly, due to the heterogeneity of the connected remote devices, FEEL faces the challenge of heterogeneous data and non-IID (Independent and Identically Distributed) data. To this end, we design a simple but elegant training scheme, namely FedOVA, to solve the heterogeneous statistical challenge brought by heterogeneous data. In this way, FedOVA first decomposes a multi-class classification problem into more straightforward binary classification problems and then combines their respective outputs using ensemble learning. In particular, the scheme can be well integrated with our communication efficient algorithm to serve FEEL. Numerical results verify the effectiveness and superiority of the proposed algorithm.