Getting a variable assignment of a Tseitin transformed formula
Given a circuit $C$ on variables $V$ which has gates $G$, the Tseitin transformation produces a formula $T(C)$ on the set of variables $V \cup G$, with the property that the formula holds iff


The value of every gate variable $g \in G$ is exactly the value of this gate given the assignment to the inputs, and
The circuit outputs TRUE.


You can prove property (1) by induction. Property (2) holds since you add a specific clause stating that the top gate in the circuit is TRUE. Given this, there is an assignment causing the circuit $C$ to output true iff $T(C)$ is satisfiable, and moreover you can read this assignment from a satisfying assignment to $T(C)$ &mdash; just ignore the gate variables.

The crucial point is that in any satisfying assignment of $T(C)$, the value of the gate variables is completely determined by the value of the input variables.