I live in Lynchburg, Virginia but drive up 29 often enough (in the bottom right of the box, and most of that area is apparently in the silent zone. The only things I've noticed are FM radio stations tend to fade more in that region and my phone never seems to work, but the roads are sort of embedded in the hills so this would seemingly be the case anywhere.
As far as enforcement goes, I saw this article on NPR several years back. They have a map on that page with a circle in it that represents where the rules are more "actively" enforced (example: man drives around in truck, looking for radio signals, asking people to turn off WIFI or whatever). When I put on my radio knowledge hat, I noticed in the article the things they appear to be blocking the most are "microwave" transmissions. (You can never escape the super low frequency signals they use to talk to submarines or the radios some hobbyists bounce off the ionosphere). I'm willing to bet there are areas of Alaska more quiet on the RF spectrum than a place several hours from DC.
It's important to remember this area is pretty hilly and rural. Even if there wasn't a quiet zone here, there just isn't much merit for lots of high-power equipment.
So will it affect your travels here? No. You likely won't notice unless you go to the receiver itself (which I hear has a visitors office) but that's not a place you would accidentally stumble into.