I’ve seen several web pages that explain how to calculate vaccine efficacy, but most of them explain it in terms of a formula rather than explaining what the formula means.
Here’s the simplest conceptual explanation I’ve seen:
a VE of 90% indicates […] a 90% reduction from the number of cases you would expect if they [had] not been vaccinated.
(From a CDC page.)
A NYT lesson plan gives some concrete numbers, which I think helps:
The Pfizer study included 43,661 people, of whom half were in the control/placebo group and half were in the group who received two doses of the vaccine.
“In the placebo group […] 162 [people] became infected with the coronavirus and showed symptoms.” In the group that received the vaccine, only 8 people became infected and symptomatic. (For the rest of this post, I’m just going to say “infected” as shorthand; but we don’t know for sure whether people who receive the vaccine can be asymptomatic carriers.)
The placebo group and the vaccine group were essentially the same size in this case, so you can think about those numbers as the vaccine preventing 154 of the infections that would have happened without it. (That’s 162 infections in the control group, minus 8 infections in the vaccine group.) (Probably if the vaccine group hadn’t received the vaccine, they wouldn’t have had exactly the same number of infections as the control group; but for the purposes of this calculation, we assume that they would have.)
If you divide that 154 number (the number of infections that were prevented by the vaccine) by 162 (the number of infections in the control group), you get 95.06%, which is the efficacy of the Pfizer vaccine. Or to put that another way, about 95% of the people who would have been infected without the vaccine were kept from being infected by the vaccine.
(If the control group and the vaccine group are different sizes, then you have to do a slightly more complicated calculation: first you determine the percentage of the control group that were infected (0.7421% in the Pfizer case); then you determine the percentage of the vaccine group that were infected (0.0366% in the Pfizer case; then you subtract the latter from the former (0.7055%), and divide by the former (95.06%). That’s the real formula; but when the two groups are the same size, various things cancel out such that you can skip some steps if you want to, and you can use the numbers of people infected instead of the percentages of people infected. The efficacy number comes out the same either way.)
For more details, see Wikipedia’s Vaccine efficacy entry.