The "paper" used in paper piecing is really a lightweight stabilizer. I like "Tear Easy" by Sulky. It is pretty translucent and really does tear easily. Do NOT buy any kind which is "fusible". The pattern is traced onto this paper. A single object (like an elephant) may require several pieced sections, which are then joined to make the final object. After the pattern is traced, small scraps of fabric are cut and sewed on sequentially. Some objects may require you to sew from the middle outwards, others from one side to the other.
Let's see how this works:
Pin the first piece in place anywhere in the marked area (#1 or A or however it is indicated). Now cut the second piece. Then, this is where it gets a little tricky. The second piece is placed right-side up, and at least 1/4" past the seam line to be sewn. You have to kind of lift the stabilizer and slide the fabric into place. It might be easier to understand from these photos:
Then of course you will turn the whole thing over in preparation for sewing on the next piece. It should look like this:
Now you have to tear off the stabilizer pieces. Start at one edge (or maybe several. Often I pull the paper off in the reverse order of how they were sewed on; the last piece sewn will always be on an edge). Fold the paper back along a seam line, crease it, and then it should tear off fairly easily. Your seam ripper and a tweezers help to get the little bits that may stick where several seams meet.
Finally, here are two finished spokes: