You are right that Shakespeare uses nature in his play with some dramatic effects. That he does this should not be surprising, as the use of nature has been a part of literature for a very long time.
Perhaps, an example would be helpful. In Shakespeare's Julius Caesar, there are many references to nature. We see this in almost every important juncture of the play.
We see a noteworthy storm before the murder of Caesar. This shows that something tumultuous is about to happen. These are seen as portents from a classical point of view, of which Shakespeare knows. Also we see many things that go against nature - again these are portents that signify that not all is well. So, we see owls in the middle of the day, and lions who walk around Rome casually.
In short, odd things in nature or extreme things in nature are used to show uneasiness.