as someone who has recently stained a fence: what I see is 2 different methods. For all the planks, they used a gold-ish colored stain. You can apply that with a wide (5" or so) stain brush you can buy at the big home improvement stores or ace hardware for like $10. it's really easy, just paint it on until you're satisfied.

The harder part here is stage 2: 3 boards were covered in what looks like gray chalk paint and sanded down. they may have added a little black to some parts of the board, too. 4 boards appear to have been dry brushed in black/dark gray chalk paint. All were sanded unevenly, probably with like 40-60 grit sandpaper, to make it look weathered.

good luck!

The best tip is to follow the instructions as stated on the stain can. You can watch some YouTube videos or read a few articles to get familiar with the process but the manufacturers instruction are what you need to follow. You should try a test piece too and that may even be in the instructions.

As for the woodworking part, cedar fence pickets are extremely rough and likely will have a bow in them. For the look in the picture, you'll need a rough sanding to remove the burs/splinters and some jointing to remove the bow. I wouldn't be surprised if there was a 1/2 piece of plywood on the back with a 2" offset so you don't see the backing and give it some more of that floating look. Makes sense too as you can screw in through the back to secure the stained wood up front.

Practice on a crap of the same type of wood you're using after you trim the pieces. Stain goes a long way, so get the small container.