In this tutorial paper, we systematically present the fundamental operating principles and analytical details of the discrete Fourier transform based signal processing techniques for the TDM MIMO FMCW millimeter-wave (mmWave) automotive radars. The mmWave radars provide a key sensing capability to support safety features of the conventional and autonomous vehicles. Automotive radar sensors are used to detect presence and location of the objects of interest to derive comprehensive and accurate knowledge of road conditions and surrounding environments. Automotive radars are subjected to an increasing demand to provide high-resolution measurements in the range-Doppler-azimuth-elevation domains. Therefore, the current state-of-the-art automotive radars commonly employ MIMO technologies, resulting in a large block of multidimensional data to process in real time through a long chain of signal processing algorithms. Detailed coverage on the fundamental radar signal processing techniques are scattered in a large body of literature for classical radar systems. Currently, there is no tutorial available that covers the technical details of the latest TDM MIMO FMCW radar technology, making it extremely hard for new researchers in the field of automotive mmWave radars. This paper contains sufficient technical details to serve as a tutorial. It can help lay a solid analytical foundation to facilitate exploration and development of advanced radar signal processing techniques for automotive applications. The algorithmic details and analytical results presented in this paper can be readily applied to both real-time implementation and post-processing. Simulation and experimental results are presented in this paper to validate analytical derivations and to demonstrate the capabilities of the TDM MIMO radar sensor in practical implementations.