Can anyone tell me why the answer is C.

I think because the lines are parallel (since, alternate angles) so the arc length will be the same. Does anyone have a better reason as to why the answer must be C ?

That itself would be enough. Notice that angles XYW and YWZ are the same.

But the angles are formed on the circumference and not the center

I am not sure how that matters as long as the lines are parallel.

I kind of thought that, if we were to move XY till XY is essentially is a point, the arc length would change, but I think that is not the case here. To stop the figure from moving up or down, I thought we would need to fix it at the center. So, I thought center was important.

If the angles enclosed by the lines joining the two ends of arcs(or chords) to any point on the circumference are equal, then their length will be the same.