I'm not sure at what level of sophistication you expect this answer, but the basic theory of how tDCS works at a cellular level is that neurons oriented roughly parallel to the electric field lines suffer a (partial) depolarization of the axon hillock cell membrane when the dendrites of the neuron are oriented toward the anode of a constant electric field, and the a reverse effect (hillock membrane hyperpolarization) occurs when the dendrites are oriented toward the cathode. (Niedermeyer's Electroencephalography, 6th ed., p. 1136) And on this basic theory, not much happens if the neuron is perpendicular to the electric field. Another way of saying the same thing (but more concisely) from Lefaucheur et al.:
The primary effect of tDCS on neurons is a subthreshold shift of resting membrane potentials towards depolarization or hyperpolarization, depending on current flow direction relative to axonal orientation.
The simple theory above applies to a cell in isolation. I'm even less sure I understand your detail question "How that positive or negative current interferes with electric fields generated by neurons?" In more sophisticated approaches, the cell shapes also affect the electric field influencing nearby cells; see for instance Ye & Steiger. Ultimately, in a nonhomogenous medium like the brain, it's quite nontrivial to determine the local electric field orientation induced by some distant electrodes. To get a solution one needs to simulate the entire brain as done by Wagner et al. for instance.
To verify that tDCS basically works like this (i.e. as a modulation, as you correctly observed) one can experimentally add another stimulus that can trigger neuronal activation, for instance TMS, and measure how much of this extra stimulus is required to obtain the same effect with and without tDCS. This was done for instance in the highly-cited study of Nitsche and Paulus. They observed excitability changes of up to 40% using this method.
The basic theory doesn't quite explain everything relating to tDCS, in particular it doesn't explain why some effects persists for a time after the DC current is turned off; the extent of these effects is dependent on the current intensity and the duration of the stimulation. Nitsche and Paulus discuss some theories relating to this, mentioning that perhaps tDCS induces changes in the spontaneous discharge rate as well.