The short pieces covered in lining are stitched, fabric sides together onto the edges of the lid, with whip stitches, fairly close together, if you can manage them (a straight needle is all you need). Then you sew the ends of them to each other, one at a time, so that the brim is formed. The outside of the lid is done next, by sticking on the batting and trimming to size, cover with the outside fabric, I pin it in place temporarily, then glue the sides down to the outside of the brim, and that is where I used spring loaded clothes pegs to hold the glued edges in place.
Finally, the 28" x 2" strip of outside fabric finishes off the brim. Iron a half inch turning on both long sides, begin near one of the joins, and stitch along the brim edge first. I take a tiny stitch (just a few threads) from fabric and lining, then slide my needle about 1/8th inch along to come out again so that you can take another bite. It is like a whip stitch, but the sloaping part ends up inside, out of site, and there is only a very tiny stitch showing, every eighth of an inch.
When you have gone all around the lower edge of the brim, fold under the join and sew it directly over the 'corner' where you started, then go around the top of the brim, where it joins the top itself. Use the same stitch, and it will be reasonably hard to see, providing you only take a few threads on the needle.
If you think the pins holding the outer brim fabric will get in the way, I have sometimes used some Elmers School glue to hold it in place instead of the pins. Use one of those very fine metal tips to 'draw' a find bead of glue along both long edges, then use an iron to heat set it in place. Don't rely on that glue as a final solution, rather regard it as a form of tacking to hold in place while you get the stitching done. Hope that helps and is clearer than mud.