Generally, you can get pretty far in these problems thinking of the circuit like a classical computer problem. That is, think about what effect this circuit would have on the following 4 quibits:
$$ |00 \psi \rangle,|01 \psi \rangle,|10 \psi \rangle,|11 \psi \rangle$$
Now think about the action that the circuit will have on these states. For the first state, what gates will the circuit apply to the third quibit? It will apply no gates.
But what about the second state? The circuit will apply a V gate to Psi and then a V-dagger gate, undoing the first action.
If you examine all of these, you can see the action of the circuit.
Generally, however, you must be much more careful because quantum circuits work for all states, not just ones in the computational basis. However, even if your initial state is in a superposition of basis states, the effect of the circuit will be a to transform your initial state to a superposition of solutions.
If we asked what effect the above circuit will have on the state $$ |+0 \psi \rangle,$$ we could quickly see that it will be a superposition of the effect on states $$|00 \psi \rangle, and |10 \psi \rangle$$.
I appologize for my abuse of LaTeX. This if my first time with it.