What is the importance of the setting in Act 1 of Hamlet?

Three of the five scenes in act 1 of Hamlet are set on a "platform before the castle" of Elsinore, at night. This is the place where sentries watch for possible threats and protect the castle. It is, therefore, open to the elements, intrinsically frightening and dangerous. This is a setting which emphasizes the tense atmosphere at the start of the play. The Danish court is threatened from inside and outside and is full of plots and rumors. The audience is made acquainted with some of these in this tense, hostile environment, where the soldiers and nobles must be on their guard constantly.

The other two scenes take place in a "room of state" and in a room in the house of Polonius. The grand, though somewhat nervous, public display of the new king of Denmark is therefore contrasted both with the intimate domestic setting that follows it and with the precarious military situation which is displayed and explained in the first scene.

What is the significance of the setting in Hamlet?

The why of the setting comes from the text Shakespeare used as raw material for his play, which was a historical account of Denmark written by Saxo Grammaticus in which Saxo wrote down the oral tradition of Amleth, the tragic Danish prince. That's where the story came from, and viewers might have been familiar with the story as with other stories Shakespeare adapted.

Now, as far as the significance, that's different. Look at the smaller setting: the castle, enclosed, a tight world where everyone's around one another and always on display. Hamlet grew up there, and knows it well, yet he finds it subverted and changed through his uncle's action. The familiar and known setting is completely transformed, making this a truly weird place for him.