This is because deoxyribonucleic acid (DNA, the building block of genes and chromosomes) has two strands, much like a zipper. The "broken genes" in this situation were in a single strand of the DNA, like breaking one tooth of the zipper.
Natural cosmic radiation and chemical processes in the body cause these types of breaks all the time. You have trillions of such breaks in your body right now. However, living organisms have repair enzymes that constantly repair this kind of break, with essentially 100% accuracy, using the opposite strand of DNA as a template to fix the damaged strand.
Problems only arise when there is a break of both strands of DNA, at the same level or nearby. You can imagine that it would be much easier to repair a single tooth of a zipper than to sew it back together after both sides are torn through.
Double-strand breaks can occasionally happen when a very high-energy particle strikes the DNA directly and tears it apart. However, this is rare even with high exposures.
Most of the damage from radiation occurs because radiation particles interact with water molecules, producing hydrogen peroxide and other 'free radicals.' These chemically interact with the DNA and produce multiple single strand breaks.
Then, the race is on between the body's repair enzymes and the damaging free radicals. If single-strand breaks accumulate faster than they can be repaired, eventually two will occur at the same level or near each other, producing a double-strand DNA break. The fact that some families have a lot of cancer and others have practically none probably relates to genetic variability in the amount and effectiveness of the DNA repair enzymes.
If the ends are not widely separated, sometimes this type of break gets repaired correctly, but quite frequently it is put back together incorrectly or not at all. When this happens, the effect is variable depending on where it is in the gene. It may do nothing at all, or it may induce cell death, or it may cause cancer, or if it happens in a sperm or egg cell, it may cause a hereditary mutation.
The particularly interesting thing is that the repair enzymes are activated by increasing levels of single strand breaks, so low levels of radiation actually rev up the repair mechanism. This has been shown not only in the rat studies you mentioned, but in studies of Hiroshima survivors, Chernobyl residents, and nuclear plant workers. The rates of cancer in some of these populations that have been exposed to significant doses of radiation are two-thirds or less of normal. This effect is called 'radiation hormesis'.
It may be premature to say that human health would be improved by building more nuclear power plants, but it certainly seems that much of our fear of radiation is not only overblown but dead wrong.
For more info: Low-level Radiation & Health
-ccm