The models used to generate R-numbers have some assumptions which may or may not be valid: (1) rectangular and stationary age distribution, and (2) homogeneous mixing of the population. Thus, the result is a fairly rough estimate.
A major use is in getting a decent estimate of what percentage of a population needs to be vaccinated in order to halt the spread of a viral infection. However, this supposes 'sterilizing vaccination', i.e. vaccinated individuals are not asymptomatic carriers and spreaders of the infection. While this was the case for the smallpox vaccine, it doesn't seem to be the case with all known Covid19 vaccines, where there are many breakthrough cases (even though symptoms are reduced and hospitalization is minimal).
The size of the total epidemic varies sharply as R is changed in this way.
To sketch here:
Consider R the reproductive number of a disease (or R_e, the effective reproductive number, to be specific). This depends on both the inherent contagiousness of the disease, but also on the behavior of the population. If people choose to have fewer contacts (or bars are closed) then R decreases.
Let's say covid is spreading. We ask people to limit their contacts, and see if this stops spread.
We can think of this as trying to remove edges from the contact graph that the disease spreads on.
The contact graph becoming connected or unconnected as we remove edges is clearly percolation.
(Subject to some modeling assumptions about edges being removed at random, but these assumptions are common to both Erdos renyi graph models and SIR style compartmental models).
Make sense?