The recently discovered vanadium-based kagome metals AV3Sb5 (A = K, Rb, Cs) exhibit superconductivity at low-temperatures and charge density wave (CDW) order at high-temperatures. A prominent feature of the charge ordered state in this family is that it breaks time-reversal symmetry (TRSB), which is connected to the underlying topological nature of the band structure. In this work, a powerful combination of zero-field and high-field muon-spin rotation/relaxation is used to study the signatures of TRSB of the charge order in CsV3Sb5, as well as its anisotropic character. By tracking the temperature evolution of the in-plane and out-of-plane components of the muonspin polarization, an enhancement of the internal field width sensed by the muon-spin ensemble was observed below TTRSB = TCDW 95 K. Additional increase of the internal field width, accompanied by a change of the local field direction at the muon site from the ab-plane to the c-axis, was detected below T * 30 K. Remarkably, this two-step feature becomes well pronounced when a magnetic field of 8 T is applied along the crystallographic c−axis, thus indicating a field-induced enhancement of the electronic response at the CDW transition. These results point to a TRSB in CsV3Sb5 by charge order with an onset of 95 K, followed by an enhanced electronic response below 30 K. The observed two-step transition is discussed within the framework of different charge-order instabilities, which, in accordance with density functional theory calculations, are nearly degenerate in energy.