The reason they know the radiation wasn't from 'radon rocks' is because there are not radon rocks everywhere. If that were the case, they could just clean up the area or move the telescope to a different location and try again...
They also waited for 6 months to check their readings again, so then the earth would be on the other side of the sun, in a completely different region of space, yet when they pointed their equipment to the skies ir still detected exactly the same phenomenon. This radiation was everywhere, in every direction, and not coming from earth.
The theory predicted it would be 3K basically 'because maths'. I don't even know how they predicted it, but the simple fact is that it
had been predicted, and
then it was discovered, which is quite a marvellous triumph for theoretical physics really. Quite a lot of things were theorised before they were discovered, actually. It's why theoretical physicists get so smug about their capabilities, hah.
On and also, the cosmic background radiation perfectly matches the intensity spectrum of a blackbody object. Which is a fantastic sign of it being from space and being a natural phenomenon from the purest days of the universe. That may be beyond the scope of A-level physics though.