The excitation of the spin degrees of freedom of an adsorbed atom by tunneling electrons is computed using a strong coupling theory. The excitation process is shown to be a sudden switch between the initial state determined by the environmental anisotropy to an intermediate state given by the coupling to the tunnelling electron. This explains the observed large inelastic currents. Application is presented for Fe and Mn adsorbates on CuN monolayers on Cu(100). First-principles calculations show the dominance of one collisional channel, leading to a quantitative agreement with the experiment.PACS numbers: 68.37. Ef, The way electrons flow through atomic contacts has important fundamental and technological implications [1]. Electronic transport is a quantal process in which charge, spin and vibrational degrees of freedom are entangled leading to problems of intrinsic fundamental interest. Technologically, the quest for minutarization is pushing the limits of devices to the atomic scale, where the above transport properties will determine the actual device functionalities. An important issue is the appearance of inelastic effects where energy is taken from the electron flow into the different degrees of freedom of the system. Inelasticities lead to new regimes of transport that contain relevant information on the atomic contact and have been thus used to develop single atom and molecule spectroscopies [2,3,4].Inelastic electron tunneling spectroscopy (IETS) where electrons excite vibrations leading to conductance steps at certain voltage thresholds [2] has been extensively studied in the last years [5,6,7,8,9]. The inelastic change in conductance is within a few percent of the elastic conductance, mainly due the smallness of the electron-vibration coupling [10,11]. Recently, Heinrich and co-workers have been able to develop a spin-resolved spectroscopy using an STM [4,12,13,14]. In magnetic IETS [4], the tunneling electron yields energy to the spin of an adsorbed magnetic atom and in this way changes its orientation by overcoming the magnetic anisotropy barrier of the atom on the surface. Magnetic transitions in the meV range could be observed in adsorbates partly decoupled from a metal substrate [4,12,13,14,15]. As in vibrational IETS, the conductance presents a step at the energy threshold however the changes in conductance at inelastic threshold can reach several hundreds percent. This is at odds with previous treaments [13,16,17] where first-order perturbation theory is used.In this letter, we present an all-order theory of the spin transitions IETS and apply it to the cases of Fe and Mn adsorbates on a CuN monolayer on Cu, experimentally studied in Refs. [13,14]. We compute the relative weights of both elastic and inelastic channels, leading to a quantitative account of the inelastic currents in the experimental observations. The theory reveals the nature of the inelastic transitions and explains the extremely large inelastic currents in these magnetic systems.The general idea of our approach is the following. ...