The cross section of the exclusive $eta$ electroproduction reaction $epto e^prime p^prime eta$ was measured at Jefferson Lab with a 5.75-GeV electron beam and the CLAS detector. Differential cross sections $d^4sigma/dtdQ^2dx_Bdphi_eta$ and structure functions $sigma_U = sigma_T+epsilonsigma_L, sigma_{TT}$ and $sigma_{LT}$, as functions of $t$ were obtained over a wide range of $Q^2$ and $x_B$. The $eta$ structure functions are compared with those previously measured for $pi^0$ at the same kinematics. At low $t$, both $pi^0$ and $eta$ are described reasonably well by generalized parton distributions (GPDs) in which chiral-odd transversity GPDs are dominant. The $pi^0$ and $eta$ data, when taken together, can facilitate the flavor decomposition of the transversity GPDs.