Graphene, a two-dimensional monatomic layer of carbon material, shows its weak absorptance of only about 2.3% in the visible and near infrared region. This dramatically limits its application in graphene-based solar cells and photon detection. In this letter, we propose a metal-dielectric periodic nanostructure to enhance the absorptance of graphene based on the magnetic resonances. Using finite-element method, we numerically simulated the effect of the structural parameters of the nanostructure on the absorptance of graphene layer. Our calculations show that the absorptance of graphene can be enhanced by about 4 times with specific structural parameters. Inductor-capacitor (LC) circuit model is applied to reveal the mechanism behind the absorption enhancement of graphene. The LC model succeeds in predicting the magnetic resonances, which are independent of the presence of the graphene. Our proposed nanostructure, which is easy for practical fabrication, is expected to have potential applications in the design of novel graphene-based optical and optoelectronic devices.