Generally, there are two power leads to car radios: switched with the ignition, and always live (for the preset memory and clock and whatnot). That's basically what the Audiovox instruction is showing. I think they show the wire going directly to the battery to emphasize that it's not switched at all; I certainly would want a fuse, at the least, on that line!
If you connect the switched line to unswitched power rather than switched power, the radio will operate regardless of the key position. As Kevin suggested, it makes some sense to just hook it up to the house electrical system. That's the way Coachmen wired things in my motorhome, in fact. Some motorhomes have a switch to select which electrical system powers the radio, but I'm not sure I really see too much utility in that option.