Helioseismological observations of the internal dynamics of the Sun during the last two solar activity cycles make it possible to trace the development of solar dynamo processes throughout the depth of the convective zone and to link them with models of solar cycles. Observational data obtained from the SoHO (1996-2010) and SDO (2010-2020) spacecraft represent measurements of the internal differential rotation, meridional circulation, and thermodynamic parameters. The structure and dynamics of zonal and meridional plasma flows reveal the processes of generation and transfer of magnetic fields inside the Sun. The data analysis shows that active latitudes and regions of a strong polar field on the Sun's surface coincide with regions of deceleration of zonal currents ("torsional oscillations"). The observed structure of zonal flows and their latitudinal and radial migration in deep layers of the convective zone correspond to dynamo waves predicted by dynamo theories and numerical MHD models. The data indicate that the development of a new solar cycle begins at about 60 degrees latitude at the base of the convective zone during the maximum of the previous cycle. Then, the process of magnetic field migration to the Sun's surface is divided into two branches: fast (in 1-2 years) migration in the high-latitude zone and slow migration at middle and low latitudes for ~ 10 years. The subsurface rotational shear layer ("leptocline") plays a key role in the formation of the magnetic "butterfly diagram". Both the zonal flows (“torsional oscillations”) and the meridional circulation reveal the 22-year pattern of the "extended" solar cycle, initially discovered from observations of Doppler velocities and the structure of the solar corona. A self-consistent MHD model of the solar dynamo developed in the mean-field theory framework is in good qualitative and quantitative agreement with the helioseismic observations. The model shows that the observed variations of the solar dynamics are associated with a magnetic field effect on convective heat transfer and the corresponding modulation of the meridional circulation. The model explains why the solar minimum polar field predicts the next sunspot maximum and points to new possibilities for predicting solar cycles from helioseismological data.