It's advisable to have read the "Normals" chapter first.
"Smoothing groups" are a way of defining areas on a mesh that should be rendered with smooth shading. Usually, numbers are used to define the group membership per polygon. One polygon can have one or several numbers and thus one or several groups. The idea behind assigning several groups to one polygon is to have a way of connecting separate groups at specific points. Let's have a look at some examples.
simple smoothing groups
Here we can see a simple smoothing group assignment where two halves of a sphere have two different smoothing groups. As expected, we have a hard non-smoothed border between both smoothing groups.
overlapping smoothing groups
In this example, we add an additional group in between the existing groups and assign both smoothing groups IDs. This way we create a connection between the groups and we get a smooth transition as you can see on the right.