Each variable has two states: true or false. In -SAT, both states matter deeply. I'll write x for "x is true" and NOT x for "x is false." Every variable creates two nodes in the graph you will build: one for x being true, one for x being false.
If you pick true for x, you cannot also pick NOT x. That seems obvious but it is the constraint that drives everything. The graph structure will enforce this constraint automatically. You will see how SCCs make it work.