Hmmm.... I definitely wouldn't say you're a
bad writer however I must say your style is a bit unorthodox. There were a few issues that I had with the piece although I think it might be a cute concept.
1) It's very short and seems very rushed. The main character's thoughts seem pretty frantic as well (understandable, but it only adds to the whole rushed feel)
2) I would say that 90 percent of this piece is Aang or Katara's thoughts. In addition, there is no real transition between the two points of view so it's a bit jarring.
3) I think that author's notes shouldn't be in the actual story but after or before. When you have author's notes in the story, it interrupts the flow.
4) I don't think the background should be established in what appears to be an author's note format. Instead of saying
"after their fight and remember this is in their heads so they aren't talking they are just walking by each other in complete and utter silence
"
You could establish this background information in the actual piece. I'm taking some creative liberties (I'm assuming this story takes place after The Cave of Two Lovers episode) however I feel something more in the mood of:
Aang's eyes seemed glued to the ground as he walked in silence. Katara walked close by but she too was lost in thought. Aang couldn't bring himself to look up at his companion, fearing what expression might be on her face. He cursed his big mouth as the two continued on in awkward silence.
I dunno, something of that sort. It provides background information but still engages the reader. I think you could add more detail and flesh out the story further. To me, though it's a good start, I feel that it's only a first draft somewhat. That's just my opinion, though