Here is the core idea that makes everything work: the formula is satisfiable if and only if no variable and its negation are in the same SCC. If x and NOT x share an SCC, you can reach NOT x through some path starting at x. That means picking x to be true forces NOT x to also be true, which is impossible.
If they are in different SCCs, you can find a valid assignment. The SCC structure tells you exactly which assignments are possible. I'll show you how to construct one next.