The WTS sometimes turns the "stumbling" concept into a manipulative weapon.
The Biblical concept of "stumbling" is that if you know someone else has a weakness you shouldn't do something that will cause him to cave in to that weakness. For example, if you have a friend who is trying to fight alcohol addiction you shouldn't invite him to go somewhere where everyone else will be imbibing.
I think it's perfectly legitimate for a friend to tell you that he has a drinking problem and that he'd really appreciate it if you would refrain from serving alcohol while he's over. But I think it crosses a line if the friend tries to manipulate you into swearing off alcohol altogether because the mere thought of you drinking booze will stumble him.
But some JWs try to get others to refrain from doing things "so you don't stumble me." This can be used to try to stop JWs from questioning anything the organization teaches or does or engaging in any behavior that will upset them. Essentially what such people are saying is, "My faith is so weak that if it will be your fault if I falter." Yet such people often see themselves as the strong ones. They think they are keeping you from error.
Following the WTS' example, such people are trying to keep others in line through guilt trips.