I did not understand this scene. It was presented in a way that somehow communicated that saving the dying hedgehog somehow drove away the attacking spiders. They made it look like if the hedgehog died, the spiders would have made a short meal of Radagast. What just happened?
Radagast was using non-magical means to try and save the hedgehog. He then realises the entire sickness of the forest as a whole (including the spiders) is magical in origin, and dark magic at that. He realises that his magic will drive away the spiders and allow him to cure the hedgehog. When he uses his staff, then the mere use of his magic scares off the spiders.
He could most likely have used any spell, but judging by Radagast's temperament saving the hedgehog was at least as important as saving himself.
I was intrigued by this scene and have an other perspective. I believe that some (or a lot) of Radagast's power comes through his creatures/friends/pets. The dying hedgehog was first, a sign that something was wrong, then after realizing it was dark magic, a sign that something/someone was attacking him. If you review the scene carefully you'll find out that Radagast almost died while curing the hedhehog. My view is that should the hedhehog had died, he would have died as well.
My interpretation of this scene is that the hedgehog was cursed as a lure to draw the spiders to Rhosgobel. I don't know why it would be necessary, but it's the only reason why I could think of that scene taking place. The spiders then apparently left because Radagast would have been able to fight them off or something like that.
Or perhaps, they were told to bring him to Dol Guldur if they didn't kill him first, so that Angmar could kill him instead. Just a thought, though these are both rather weak theories.
The spell Radagast uses to drive away the spiders is very powerful magic that was used to combat the spiders whilst also curing Sebastian. Sebastian had been cursed by dark magic in an attempt to make Radagast lower his guard as it was playing on his gentle nature and concern for the animals. As the spiders attacked it became clear to Radagast that it was dark magic that had killed Sebastian and so he used the stone within his staff to draw Sebastian's very life essence from his body and used the dark magic within to drive back the spiders. After this was done Radagast returned the now clean life essence to Sebastian restoring the hedgehog to life. This was possible because the dark magic that had killed Sebastian was redirected against the spiders and so no longer focused on Sebastian.