Sleep staging is important in sleep research since it is the basis for sleep evaluation and disease diagnosis. Related works have acquired many desirable outcomes. However, most of current studies focus on time-domain or frequency-domain measures as classification features using single or very few channels, which only obtain the local features but ignore the global information exchanging between different brain regions. Meanwhile, brain functional connectivity is considered to be closely related to brain activity and can be used to study the interaction relationship between brain areas. To explore the electroencephalography (EEG)-based brain mechanisms of sleep stages through functional connectivity, especially from different frequency bands, we applied phase-locked value (PLV) to build the functional connectivity network and analyze the brain interaction during sleep stages for different frequency bands. Then, we performed the feature-level, decision-level and hybrid fusion methods to discuss the performance of different frequency bands for sleep stages. The results show that (1) PLV increases in the lower frequency band (delta and alpha bands) and vice versa during different stages of non-rapid eye movement (NREM); (2) alpha band shows a better discriminative ability for sleeping stages; (3) the classification accuracy of feature-level fusion (six frequency bands) reaches 96.91% and 96.14% for intra-subject and inter-subjects respectively, which outperforms decision-level and hybrid fusion methods.