It gets one about 85-90% of the way there. That's very close.Of course the better defense helped, but to argue that that explains everything is overly simplistic. There was a lot more to it.
1. Arroyo adjusted his pitching style to take advantage of the better defense, as mth123 has stated. Other Reds pitchers didn't benefit from that same defense, which supports this notion. Bailey underperformed his peripherals over those three seasons from 2009-2011. As did Harang his two years with the Reds during that period. Cueto didn't improve with the defense, until he adjusted his pitching style in 2011. And Volquez got much worse, despite the better defense. So it clearly wasn't just the defense, or else all the other starting pitchers would have benefited as well.
2. If you look closer, except for 2008 and 2011, Arroyo really didn't improve all that much. He was very similar in terms of how he compared to the rest of the league. Meaning his ERA went down as scoring went down across the league. From 2007-now, Arroyo has produced between a 105-111 ERA+ every year except for 2008 and 2011. So he hasn't improved that much, compared to the league. Scoring has just gone down everywhere.
So yeah, the Reds improved defense has helped Arroyo, but that's not the whole story, not even close.