The problems are discussed in the paper that may have a fundamental importance for understanding and estimation of possible role of solar activity in climatic variations in the past and the future. Particular attention is paid to the mechanism of solar activity influence on the climatic characteristics of the atmosphere through the tropospheric electricity. The principal causes of significant variations of the electric field in the Earth's atmosphere over polar latitudes during the events of helio-geophysical disturbances are analyzed. Those are the solar cosmic rays, protons, and electrons precipitating from the magnetosphere during geomagnetic disturbances and enhancement (disturbances) of magnetospheric convection during the geomagnetic storms. It is proved that this mechanism works efficiently in the high-latitude regions, favoring the formation of extra cloudiness in the regions with sufficient moisture content. We also present the estimations of temperature, pressure and radiation balance variations in the high-latitude regions, in which water vapor phase state changes due to the influence of helio-geophysical factors. It is shown that the greatest and unique influence of solar activity on the state of climatic system is observed in the absence of incoming radiation (high-latitude regions in winter), as in this case any cloudiness causes a decrease of the energy losses by the climatic system. In the framework of the model and mechanism of solar-terrestrial connections proposed, the results of investigation of manifestation of solar flares and interplanetary magnetic field structure in the thermobaric characteristics of the troposphere are systematized and generalized.