1. brainish apprehension
Definition: brain sick belief
The significance of this phrase contributes to the effect of Hamlet's gradual insanity and helps to define how he has reacted to the events that have caused him to react the way that he has. This phrase has also shed light on how the people around Hamlet now perceive him, especially now that he has outright killed Polonius, yet still no one understands the truth about King Hamlet's death.
2. "Where tis so, th' offender's scourge is weighed, but never the offense."
definition: Where the people love with their eyes instead of their reason, they judge with punishment rather than crime.
This phrase contributes to the tone of the passage in that the question is what to do with Hamlet now that he has killed Polonius. Claudius contemplates what he should do with Hamlet, and considers other people's opinions and how they will view the situation. He describes their reactions in that their opinions matter, since he should speak for the people, but he fears that they may care more about the punishment of Hamlet, rather than the evil that he has done.
3. "by his license Fortinbras craves the conveyance of a promised march over his kingdom"
definition: Fortinbras has been given permission to march through Denmark and is now asking for an escort.
This adds to the effect and forseeing drama and anticipation of the upcoming war and also questions Claudius's loyalty to Denmark in that he would turn his back on his country and let the enemy inside.
4. "Will not debate the question of this straw"
definition: are not enough to pay for settling this trifling quarrel
This adds to the tone of the passage and adds to the effect of the war stricken Denmark in that the anticipation of the war being waged on Denmark is ever growing. This also reveals Hamlet's loyalty to his country in that he defends his country's honor.
This words helps the tone of this particular part of the passage in that it shows Hamlet's disgust of the war and helps the reader to better understand the war by his disgust of it being centered around the amount of wealth and peace that one might have. He is also projecting his opinion that this is not a good enough reason for anyone to die for, in other words, the war and its cause is not worth fighting for.