Altruism in viscous (asexual) populations
The $sR$ that your looking at is the average relatedness of the next generation.  This assumes that the new immigrants into the the population are completely unrelated. So if the population is completely viscous ($s=1$) the average relatedness of the next generation equals that of the current generation.  On the other hand, if the population is not viscous ($s=0$) the next generation would be entirely unrelated because it would only be made up of unrelated immigrants.

The $s(b-c)$ is the number if individuals that are displaced by new offspring.  Again if all of the new generation sticks around ($s=1$) then the number of displaced individuals equals the number of new individuals.  And if all new progeny leave ($s=0$), there are no displaced individuals.