  We present a microscopic theory to give a physical picture of the formation
of quantum anomalous Hall (QAH) effect in graphene due to a joint effect of
Rashba spin-orbit coupling $\lambda_R$ and exchange field $M$. Based on a
continuum model at valley $K$ or $K'$, we show that there exist two distinct
physical origins of QAH effect at two different limits. For $M/\lambda_R\gg1$,
the quantization of Hall conductance in the absence of Landau-level
quantization can be regarded as a summation of the topological charges carried
by Skyrmions from real spin textures and Merons from \emph{AB} sublattice
pseudo-spin textures; while for $\lambda_R/M\gg1$, the four-band low-energy
model Hamiltonian is reduced to a two-band extended Haldane's model, giving
rise to a nonzero Chern number $\mathcal{C}=1$ at either $K$ or $K'$. In the
presence of staggered \emph{AB} sublattice potential $U$, a topological phase
transition occurs at $U=M$ from a QAH phase to a quantum valley-Hall phase. We
further find that the band gap responses at $K$ and $K'$ are different when
$\lambda_R$, $M$, and $U$ are simultaneously considered. We also show that the
QAH phase is robust against weak intrinsic spin-orbit coupling $\lambda_{SO}$,
and it transitions a trivial phase when
$\lambda_{SO}>(\sqrt{M^2+\lambda^2_R}+M)/2$. Moreover, we use a tight-binding
model to reproduce the ab-initio method obtained band structures through doping
magnetic atoms on $3\times3$ and $4\times4$ supercells of graphene, and explain
the physical mechanisms of opening a nontrivial bulk gap to realize the QAH
effect in different supercells of graphene.
