Why does voltage cancel out if a circuit is going through a magnetic field? 
The answer says that there is no current as there is no voltage, but why is there no voltage even though they cut field lines? 
EDIT; I found the answers, but still not really understanding it

 A: When a conducting wire is swinging in magnetic field, induced electromotive force is generated by the change of  magnetic flux. This is for a single wire.
On the other hand, this figure seems to fold the wire. The folded wire is swinging as a single wire with 2 wires. So, the induced electromotive force is generated with same direction for 2 wires, simultaneously. I think that this is the reason of zero voltage.
If I had misunderstood, I apologize.
A: First, "cutting field lines" is an outdated concept(Faraday) regarding induction. But this is not significant here.
Swinging the folded wire doesn't induce a voltage in it because the magnetic flux enclosed by the wire is zero. If you look at it from the viewpoint of the Lorentz force on the conduction electrons moving with velocity $\vec v$ in the wire $F=e\vec v×\vec B$, there is no net EMF induced in the wire because in half of the wire $F$ is positive and in the other half negative.
A: If you are happy with the idea that a single wire doesn't have any current flow, then the following diagram might explain why you still get no current with two wires.
Note that there is a voltage difference betwen the two ends of the wire/tape/tapes, but it doesn't produce a current as there is no complete circuit (similarly, a battery has a voltage difference between its two ends - but no current flows).

