We have deferred several mysteries and postponed some explanations of the terms Sage uses to describe certain objects. This is because Sage is a comprehensive tool that you can use throughout your mathematical career, and Sage assumes you already know a lot of linear algebra. Do not worry, you already know some linear algebra and will very soon will know a whole lot more. And we know enough now that we can now solve one mystery. How can we create all of the solutions to a linear system of equations when the solution set is infinite?
Theorem VFSLS described elements of a solution set as a single vector $\vect{c}$, plus linear combinations of vectors $\vectorlist{u}{n-r}$. The vectors $\vect{u}_j$ are identical to the vectors $\vect{z}_j$ in the description of the spanning set of a null space in Theorem SSNS, suitably adjusted from the setting of a general system and the relevant homogeneous system for a null space. We can get the single vector $\vect{c}$ from the .solve_right() method of a coefficient matrix, once supplied with the vector of constants. The vectors $\set{\vectorlist{z}{n-r}}$ come from Sage in the basis returned by the .right_kernel(basis='pivot') method applied to the coefficient matrix. Theorem PSPHS amplifies and generalizes this discussion, making it clear that the choice of the particular particular solution $\vect{c}$ (Sage's choice, and your author's choice) is just a matter of convenience.
In our previous discussion, we used the system from Example ISSI. We will use it one more time.
So we can describe the infinitely many solutions with just five items: the specific solution, $\vect{c}$, and the four vectors of the spanning set. Boom!
As an exercise in understanding Theorem PSPHS, you might begin with soln and add a linear combination of the spanning set for the null space to create another solution (which you can check).