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.