The purpose of the paper is to gain deeper insight into microscopic formation of the Dzyaloshinskii-Moriya interaction (DMI). The paper aims at the development of the physical picture able to address apparently contradicting conclusions of recent studies concerning the location of the DMI energy in the real and reciprocal spaces as well as the relation between values of the atomic moments and the DMI strength. The main tools of our study are the first-principles calculations of the energies of the spiral magnetic states with opposite chiralities. We suggest a method of the calculation of the spiral structures with account for the spin-orbit coupling (SOC). It is based on the application of the generalized Bloch theorem and generalized Bloch functions and allows to reduce the consideration of arbitrary incommensurate spiral to small chemical unit cell. The method neglects the anisotropy in the plane orthogonal to the rotation axis of the spirals that does not influence importantly the DMI energy. For comparison, the supercell calculation with full account for the SOC is performed. The concrete calculations are performed for the Co/Pt bilayer. We consider the distribution of the DMI energy in both real and reciprocal spaces and the dependence of the DMI on the number of electrons. The results of the calculations reveal a number of energy compensations in the formation of the DMI. Thus, the partial atomic contributions as functions of the spiral wave vector q are nonmonotonic and have strongly varying slopes. However, in the total DMI energy these atom-related features compensate each other, resulting in a smooth q dependence. The reason for the peculiar form of the partial DMI contributions is a q-dependent difference in the charge distribution between q and -q spirals. The strongly q-dependent relation between atomic contributions shows that the real-space distribution of the DMI energy obtained for a selected q value cannot be considered as a general characteristic of a given material. Our study shows that it is physically most consistent to consider the electronic hybridization as a primary effect reflecting the nature of the DMI whereas the q-dependent real-space distribution of the DMI energy is a consequence of the complex processes in the electronic structure including the charge transfer process. The physical process of the DMI formation is connected with the difference in the hybridization of the Co and Pt states for q and -q spirals under the influence of the SOC and broken spatial inversion. It depends sensitively on details of the electronic structure. The calculations with constraints on the values of the Co and Pt atomic moments show that there is no direct relation between these atomic quantities and the DMI strength since the details of the electronic structure crucial for the DMI are not reflected in these integral characteristics. The application of the method to the calculation of the magnon energies in systems with DMI is briefly addressed.