The idea is basicly to hit a region only to weaken not to take. There is a key, ' w ' for example, which activates the option.
For example in border distribution you want to hit a region but if you take it, your units would be distributed to more regions and thus your defance may get weaker at certain spots.
An other example: In attrition mode you have two 8s next to a 7 of your opponent. If you hit the 7, in the current situation you most probably take that region and thats it but in the case there was a "weaken-hit" option you could weaken the 7 to 1 with the first 8 and penetrate with the second 8.