Ram pressure stripping (RPS) is an important process to affect the evolution of cluster galaxies and their surrounding environment. We present a large MUSE mosaic for ESO 137-001 and its stripped tails, and studied the detailed distributions and kinematics of the ionized gas and stars. The warm, ionized gas is detected to at least 87 kpc from the galaxy and splits into three tails. There is a clear velocity gradient roughly perpendicular to the stripping direction, which decreases along the tails and disappears beyond ∼45 kpc downstream. The velocity dispersion of the ionized gas increases to ∼80 km s−1 at ∼20 kpc downstream and stays flat beyond. The stars in the galaxy disk present a regular rotation motion, while the ionized gas is already disturbed by the ram pressure. Based on the observed velocity gradient, we construct the velocity model for the residual galactic rotation in the tails and discuss the origin and implication of its fading with distance. By comparing with theoretical studies, we interpreted the increased velocity dispersion as the result of the oscillations induced by the gas flows in the galaxy wake, which may imply an enhanced degree of turbulence there. We also compare the kinematic properties of the ionized gas and molecular gas from ALMA, which shows they are co-moving and kinematically mixed through the tails. Our study demonstrates the great potential of spatially resolved spectroscopy in probing the detailed kinematic properties of the stripped gas, which can provide important information for future simulations of RPS.