R=sR_0. So the effective reproduction number is a simple linear multiple of R_0, scaled by the proportion of people who are susceptible.
Imagine a scenario where a population is 100% susceptible. (s=0) One member is infected. Infection lasts a week. During each week, the infected person has contact with 2 other people. Each contact has a 50% chance of passing on the infection. So the chance of *not* infecting anyone during the week is (1-0.5)^2, or 25%. Thus, the chance of infecting someone is 100%-25% = 75%.
Now, imagine the same scenario, except that there is 50% herd immunity. (s=0.5). So in the same story, one of those two people was immune. So they only had a 50% chance of infecting someone.
Comparing s=0 with s=0.5, we see the chance of spreading the infection change from 75% to 50%. Oops. According to R=sR_0, we expected it to go from 75% down to 75%/2 = 37.5%.
Spot my error! :)