Bsaically, your DNA gets damaged in a variety of ways and when your DNA repair mechanisms can't fix it or don't fix it correctly, you get mutations and improper protein function because your cell is accessing corrupted data. This can lead to constantly active growth signals or inactive anti-growth signals, both of which will lead to cancer. On the flip side, DNA can be mutated into overall lack of function, which leads to cellular aging, which leads to the organism aging.
What they did in this experiment was measure in peripheral white blood cells a single type of DNA damage, the strand break, which is, as the name implies, is when one or both strands of a molecule of DNA break apart. This is a nasty type of damage as, in the case of double strand breaks, it can cause chromosomal translocations, which is where two different chunks of broken DNA are put together, which may result in improper genetic function. For example, the "always active" switch of a tumor suppressor gene may be fused to a gene coding for a protein that promotes cell proliferation, which will promote cancer. In the case of single strand breaks, improper repair of the strand can lead to new mutation on its own.
Their findings indicate that those who drank 750 ml (~3 cups) of coffee per day experienced 27% fewer strand breaks in white blood cells than those who only drank water, controlling for diet and body weight
