My interpretation is that 60% protection means that where (for example) 1% of people in the Placebo group were infected, only 0.4% of the people in the Vaccine group were infected, which is in line with what DecemberStar said.
However, I think it is possible, though totally unproven for the time being, the the vaccine also gives some protection against serious illness to the other 40%. it has been written in the media headlines that nobody in the Vaccine group for the Oxford vaccine (the one with only about 60% effectiveness in the larger vaccine trial arm) ended up seriously ill or in hospital. That could just mean that so few people were infected within the Vaccine group, that by pure chance, none of them became seriously ill. However, it could also mean that some or even many of those who were vaccinated but still became ill were helped by the vaccine in that they only got mild illness instead of severe illness. My interpretation of the headlines is that on the latter point, we just don’t know yet, but I believe that as large numbers of people to start to get the Pfizer vaccine (Which actually is supposed to have over 90% effectiveness), we will find out a lot more about that one in the coming months. We will then learn much more about the Oxford vaccine in the months after it gets approved (if it gets approved).
I do also know that if the Oxford/AstraZeneca vaccine were approved and then offered to me tomorrow but I had to decide “yes” or “no” immediately, then I would definitely get it. The data are far from perfect, but the science of how vaccines work is sufficiently robust that I believe it would meaningfully lower my risk of becoming seriously ill, should I contract COVID, and I also believe that the risk from the vaccine itself would be much lower than the risk from the virus.
If the Pfizer vaccine or Moderna vaccine were offered, then I would agree even more enthusiastically. But I don’t expect there to be enough doses for me to be offered either of those before probably 2022.
Finally it is critically important to watch for news on whether a vaccinated person can still acquire and pass on an infection asymptomatically. If they can, then getting vaccinated must not be taken as a license to stop with the social distancing measures, less the vaccinated person become a modern-day Typhoid Mary.