The answer is, it's simple.
A pilot flying an instrument approach is concentrating not on the view out the front, but on the gauges on the panel. When he reaches the point in the approach known as "decision height", he looks outside, and if he sees the runway, he lands on it. With two pilots, the one not flying looks outside, and tracks the runway as the approach continues.
In this case, two things got in the way. They flew through a cloud close to decision height, which hid the runway they had been watching on the way in. Second, when they got out of the cloud, they saw the runway at Ellsworth AFB, which was in the same direction - and thus had the same runway number painted on the end - and not very far away from the runway at Rapid City. Thus, they thought that was the runway they needed, and they landed on it. The tower at Ellsworth AFB couldn't get word to them in time, since there's not a lot of distance between decision height and the runway on an ILS, and they weren't on that tower's radio frequency anyway (since they were on the Rapid City tower frequency).
This is not unheard of. When I was doing my primary flight training at Ellington Field (ex-Ellington AFB, and still in use as an Air National Guard base), a USAF student landed a T-38 jet at La Porte Municipal Airport under almost identical circumstances. It's an easy mistake to make.