You wouldn't expect the screening to affect things much in the "all other causes" part of the table, so we should focus on the breast cancer columns. The numbers of deaths aren't directly comparable, since the numbers of people incurring the deaths vary from one treatment category to another, so the key table entries are the death rates. The only fair comparison in this column is the death rate for the entire treatment group, 1.3 per 1,000 women, versus the death rate in the control group, 2.0, a comparison that shows a 35% reduction in death rate from breast cancer caused by membership in the treatment group ((1.3 - 2.0)/2.0 = 35%.
You can talk validly about causation because the randomization of 62,000 women into treatment and control groups was overwhelmingly likely to have balanced all the potential confounding factors (PCFs) quite well. Notice that it isn't fair to compare the 1.1 or the 1.5 to the 2.0 in the death rate column; for example, the 2.0 people differ from the 1.1 people in more ways than just being offered the screening - not only were the 1.1 people offered it, but they also chose to accept it (more on this below). All we can say for sure is that if all 31,000 women in the treatment group had taken the experimenters up on the offer to get screened, the death rate in that group would have been somewhere between 1.1 and 1.3 per 1,000 women.