Satin stitch applique is done with a close zig-zag stitch. There are several wonderful books on applique to give you detailed information on the technique, so this just touches basics and hopes to give you some ideas and get you to experiment.
Use a presser foot that has space under the foot for the build up of stitching you get with applique.
Use thread in the top for the effect you like. I like rayon threads like Sulky, Madeira, or acrylic New Home Embroidery threads because of their high sheen.
Use a thinner thread in the bobbin. 60 weight machine embroidery thread is nice because you can get a lot on the bobbin, and there's no build up underneath that can impede feeding. You may use white for all stitching, or match your top thread. If I'm doing a lot of tapering and straight stitching in the design, I use a color closer to my top thread.
Select the appropriate stitch and adjust the density of the stitching (stitch length) so that you're stitching closely, but the fabric moves under the presser foot. If it doesn't move, lengthen your stitch. Reduce your top thread tension if you see little knobs on the edges of your satin stitch.
Fuse your applique piece to the backing. Use a stabilizer under the backing fabric. Stitch around the applique with the satin stitch.
Change color of thread and stitch over the satin stitching with a stitch that looks like the drawing. It should be wide enough to go over the satin stitching on both sides and have the straight part on the outside. If you have a mirror function, you may need to use that, depending on your stitch. Try different stitch lengths to see what you like best.
For an additional touch: hand whip stitch another color or different decorative thread around the bars as shown below in the closeup of my kitty's pillow.