That the firing rates are proportional to the input population sizes is called the _law of mass action_.
It makes sense the the firing rate of infection will be proportional to \\(S \cdot I\\).
(EDIT: this is only true to a first approximation, for large population sizes. The refinement is described in comment 14.)
Before an epidemic has begun, \\(I(t)\\) is small, so infections occur at a low rate. As \\(I\\) begins to increase, the rate of infection increases proportionally. This is the exponential growth at the outset.
As the epidemic progresses, \\(S\\) goes down, which has the effect of reducing the base of the exponential.
At the tail end of the epidemic, the combination of reduced values for \\(S\\) and \\(I\\) pulls the infection rate down.
These dynamics are responsible for the sigmoidal shape of the cumulative infection curve.