  We study the many-body physics in twisted bilayer graphene coupled to
periodic driving of a circularly polarized light when electron-electron
interactions are taken into account. In the limit of high driving frequency
$\Omega$, we use Floquet theory to formulate the system by an effective static
Hamiltonian truncated to the order of $\Omega^{-2}$, which consists of a
single-electron part and the screened Coulomb interaction. We numerically
simulate this effective Hamiltonian by extensive exact diagonalization in the
parameter space spanned by the twist angle and the driving strength.
Remarkably, in a wide region of the parameter space, we identify Floquet
fractional Chern insulator states in the partially filled Floquet valence
bands. We characterize these topologically ordered states by ground-state
degeneracy, spectral flow, and entanglement spectrum. In regions of the
parameter space where fractional Chern insulator states are absent, we find
topologically trivial charge density waves and band-dispersion-induced Fermi
liquids which strongly compete with fractional Chern insulator states.
