The kinematics and chemical composition of stellar populations of different ages provide crucial information on the evolution of a galaxy. In particular, isolated dwarf galaxies offer us a window into the intrinsic mechanisms that shape their observed properties. Using MUSE, we have performed a new spectroscopic investigation of IC 1613, an isolated gas-rich, star-forming dwarf galaxy of the Local Group (LG). We obtained three deep fields from which we extracted around 2000 stellar sources down to I~24. The quality of the dataset allowed us to obtain precise classifications and line-of-sight velocities for about 800 stars. Our sample includes not only Red Giant Branch (RGB) and Main Sequence (MS) stars, but also a number of probable Be and C stars. We also obtained reliable metallicities for about 300 RGB stars. This provides the largest spectroscopic sample of an isolated LG dwarf galaxy. The kinematic analysis revealed for the first time the presence of stellar rotation with high significance, finding general agreement with the rotation velocity of the neutral gas component. Examining the kinematics of stars as a function of broad age ranges, we find that the velocity dispersion increases as a function of age, with the behaviour being very clear in the outermost pointings, while the rotation-to-velocity dispersion support decreases with age. The results obtained seem to support the scenario in which the stars of a dwarf galaxy are born from a less turbulent gas over time.