You seem to have misunderstood efficacy.
The 95% efficacy figure is a derived from a ratio of the number of cases in the control group (no vaccine) to the number of cases in the vaccine group. So, if both groups have 1000 people, perhaps 100 people in the control group become infected, while only 5 people in the vaccine group become infected. The vaccine efficacy is 95%, since the number of cases in the vaccine group is 95% lower than what you saw in the no-vaccine group (note that these numbers are illustrative and not actual figures).
A 95% efficacy does not mean that 5% of people will die from the disease. It doesn't even mean that 5% of people will become infected. It means that 95% of people who would have otherwise been infected will not be. If everyone was magically vaccinated today, you would expect the number of cases to drop by 95%, and the number of deaths from the disease to drop by a similar amount. Instead of 2.5% of the population dying, it might be more like 0.13%.
Note that there will be major downstream effects since viruses spread in an exponential manner - an important number you may have heard of is called R0, which represents the average number of new people an infected person will infect. A vaccine can cause R0 to drop below 1, meaning that the number of cases will dwindle over time. It's not just the case that a vaccine will prevent infection in 95% of people who would have otherwise gotten the disease, it also drastically lowers the number of people who "would have otherwise gotten the disease", since 95% of vaccinated individuals won't spread the disease.