What do we mean when we say it today?
You might have heard Gaston belting out this line in his creepy "Kill the Beast" song in Beauty and the Beast. It goes a little something like this: Grab your pitchfork and get ready to storm a castle because if you "screw your courage to the sticking place" you'll be able to hunt down the beast.
If you've heard the song, then you've probably figured out that we use this phrase today in exactly the same way Lady Macbeth does in this scene.
We whip it out whenever we want to say something like stick to the plan. Be resolute. Stop going back and forth so much. Be brave. Most of the time, it doesn't have the "man up" subtext though. People just say this when they're talking about being dedicated to something.