The correlated insulator (CI) states and the recently discovered density wave (DW) states in magic-angle twisted bilayer graphene (TBG) have stimulated intense research interest. However, to date, the nature of these "featureless" correlated states with zero Chern numbers are still elusive and lack a characteristic experimental signature. Thus, an experimental probe to identify the characters of these featureless CI and DW states is urgently needed. In this Letter, we theoretically study the correlated insulators and density wave states at different integer and fractional fillings of the flat bands in magic-angle TBG based on extended unrestricted Hartree-Fock calculations including the Coulomb screening effects from the remote bands. We further investigate the nonlinear optical response of the various correlated states and find that the nonlinear optical conductivities can be used to identify the nature of these CI and DW states at most of the fillings. Therefore, we propose that a nonlinear optical response can serve as a promising experimental probe for unveiling the nature of the CI and DW states observed in magic-angle TBG.