Direct numerical simulations (DNSs) of purely elastic turbulence in rectilinear shear flows in a three-dimensional (3D) parallel plate channel were carried out, by which numerical databases were established. Based on the numerical databases, the present paper analyzed the structural and statistical characteristics of the elastic turbulence including flow patterns, the wall effect on the turbulent kinetic energy spectrum, and the local relationship between the flow motion and the microstructures’ behavior. Moreover, to address the underlying physical mechanism of elastic turbulence, its generation was presented in terms of the global energy budget. The results showed that the flow structures in elastic turbulence were 3D with spatial scales on the order of the geometrical characteristic length, and vortex tubes were more likely to be embedded in the regions where the polymers were strongly stretched. In addition, the patterns of microstructures’ elongation behave like a filament. From the results of the turbulent kinetic energy budget, it was found that the continuous energy releasing from the polymers into the main flow was the main source of the generation and maintenance of the elastic turbulent status.