In a manner of speaking, "a stir" is the same thing as "a ruckus," "a scene," "a commotion," or any kind of unusual social situation that causes people to react to it, by gossiping if nothing else.
So, to "cause a stir" is to create an unusual situation that gives rise to excitement and interest among others. Note, however, that the "stir" is not necessarily positive. There can be such a thing as bad publicity!
Example: "Tracy caused quite a stir when she was cursing loudly in the theater. I know she thought she'd lost her cell phone, but by the time she found it, the usher had come to kick her out because she was making such a scene!"
"Making a scene" should be thought of as meaning the same thing as "causing a stir." By using two different idioms that mean the same thing, we avoid the appearance of repetition. If I'd stopped at "to kick her out!", the reason for the usher (a person who keeps order in a movie theater) to kick Tracy out might not have remained perfectly clear.