The cross section of the exclusive $\pi^0$ and $\eta$ electroproduction reaction $ep\to e^\prime p^\prime \pi^0/\eta$ was measured at Jefferson Lab with a 5.75-GeV electron beam and the CLAS detector. Differential cross sections $d^4\sigma/dtdQ^2dx_Bd\phi$ and structure functions $\sigma_U = \sigma_T+\epsilon\sigma_L, \sigma_{TT}$ and $\sigma_{LT}$, as functions of $t$ were obtained over a wide range of $Q^2$ and $x_B$. At low $t$, both $\pi^0$ and $\eta$ are described reasonably well by Generalized Parton Distributions (GPDs) model in which chiral-odd transversity GPDs are dominant. Generalized form factors of the transversity GPDs $H_T^{\pi,\eta}$ and $\bar E_T^{\pi,\eta}$ were directly extracted from the experimental observables. The combined $\pi^0$ and $\eta$ data opens the way for the flavor decomposition of the transversity GPDs. The first ever demonstration of this decomposition was done for the transversity GPDs $H_T$ and $\bar E_T$. GPD $\bar E_T$ is connected with the density of the polarized quarks in an unpolarized nucleon in the impact parameter space. The spin density of polarized $u$ and $d$-quarks was evaluated for different values of Feynman $x$ from the GPD model tuned to described the experimental data.