It is definitely "to provide/give (...) insights on and not into. Instead of "to provide some insights on how to" I would prefer to say "to give you some insight on how to...". Note that I use insight as singular, especially since you are giving just one example!
I have one problem with the question though. I cannot imagine the context where one would construct a region. I have a feeling either to construct or region is not the ideal word to use here. If you could clarify this I can improve my answer.
So with "characterize the region" it would be:
To provide some insight on how to characterize the region, we give the following simple example.
To provide sounds rather formal to me, whereas the rest is more informal. A consistent formal way would be:
To provide some insight on how to characterize the region, consider the following example.
An informal example would be:
To get some insight on how to characterize the region, you may look at the following simple example.