We propose a new error estimation framework of nonconforming finite element methods for eigenvalue problems, and provide a detailed introduction by applying it to the weak Galerkin (WG) method as an example. To further accelerate the solving process, we propose an augmented subspace method based on the WG method for solving eigenvalue problems. The augmented subspace is built with the conforming linear finite element space defined on the coarse mesh and the eigenfunctions in the WG finite element space defined on the fine mesh. Based on this augmented subspace, solving the eigenvalue problem in the fine WG finite element space can be reduced to the solution of the linear boundary value problem in the same WG finite element space and a low dimensional eigenvalue problem in the augmented subspace. The corresponding error estimates are provided to show that the proposed augmented subspace schemes have the second order convergence rate with respect to the coarse mesh size. We demonstrate the effectiveness of our new method through a series of numerical experiments.