The first program builds a Zero-Age Main Sequence (ZAMS) stellar model with the mass and composition you have chosen for the outer boundary condition calculations. The star has presumably contracted from an interstellar cloud during the past few million years (these programs do not accurately portray that physics), during which time convection has made the chemical composition the same from center to surface.
The equations which this program solves take into account energy generated by nuclear reactions near the center of the star, the transfer of that energy by convection and radiation to the surface, the hydrostatic equation (where gas and radiation pressure balance the pull of gravity), and a gas equation of state for ideal gas, radiation, and partial degeneracy.
These equations are all simultaneously satisfied when four boundary conditions are satisfied, which is what the program solves for: the central temperature Tc and density rhoc, and the surface temperature Teff and luminosity L. In order to do this, you the user must guess reasonable values for each of these quantities. Then the program solves the above equations starting at the star's center, and again at the star's surface.
The solution of the equations will provide T(m) and rho(m) (the temperature and density of the gas a distance from the star's center such that mass m is contained inside), and L(m) and r(m) (the luminosity produced and the distance from the center). These trial solutions are plotted on the screen, each function in a different color.
You will notice that in the middle of the star, the solutions will not agree, but rather, a jump in each quantity is seen. This is not physical, and results from you having guessed the wrong conditions at the center and surface. The program can improve your guesses, which it does, and tries again when you call up the next picture. Now you will see that the jumps in physical quantities get smaller with each subsequent improvement, until there is a smooth flow of each across that fitting point.
When the program has "converged" a model, it now knows the correct values of the four variables you guessed at the start. Thus the characteristics of any star are determined, once the mass and composition of the model are chosen.