New Powers
Peter realizes that he has the powers of a spider and begins to climb the walls of buildings in New York. Once he figures out he can shoot his web from his wrists, he gives it a try with great excitement before crashing into a building. The imagery reveals how incredible the fantasy of having super-powers come to life is. Peter "takes them for a spin" like anyone would, but the crashing into the wall symbolizes the reality of having to understand what to do with the powers and how to use them as ultimately they gifts that also bear the weight of life and death.
Heightened Senses
Peter is attacked by Flash at school and we watch as everything slows down: a paper airplane, a fly, and Flash's fist before connecting with Peter's face. The imagery shows us Peter's point of view with his new powers, and allows us to experience what it might be like to be in his position from a first person perspective. This lets us as the audience have as much fun as Peter has in the scene.
Revenge
Peter hunts down the man who shot and killed his uncle, and when given the chance to save the guy (who falls out of a window), Peter doesn't do anything. The man dies and the imagery reveals that Peter is still a human being, full of rage and pain which misguides his abilities in this moment.
Fog
The Green Goblin has paralyzed Spider-Man and as he awakens we see a fog around him. The imagery connects quite plainly the reality that Peter, as Spider-Man is knocked out and now in a different, uncertain place than he was before. All of this connects to Norman Osborn, who himself is living in a haze as his alter-ego, the Green Goblin.
