The Cards were in a no-win situation here.
To keep Horton, they likely would have needed to make him the HC. It's not like they were going to find a solid HC candidate by saying right off the bat, "You can come in, but you can't replace the DC."
But Arizona's biggest need is finding a starting QB and being able to develop him, and Horton simply can't do that. I imagine when he was interviewed, he was asked about his possible choices for OC and how he would develop Kolb or whoever they went with. I guess by the way things played out, they didn't rank his answer as their top choice.
Arians has been known to be good with QBs. He was Peyton Manning's first QB coach in the NFL, was the OC in Pittsburgh when the Steelers starting passing more (and Roethlisberger put up bigger numbers), and oversaw Luck last year. So I imagine that played heavily in his favor when Arizona was making their decision.
In the end, I think they hope that the talent Arizona has on D will make up for whatever schematic changes do to the D, while at the same time getting a guy who can actually make their offense work. I think the goal is to change the team where it has a solid D and an improving O rather than an outstanding D and an absolutely horrific O.
Super Bowl XLVIII Champions