If you click download from the steam browser, the plane will load automatically
If you use normal browsers but don't want to end up with a bunch of .splane files, you can copy the aircraft link, and in the designer press ctrl+L

When you download a plane, you're actually downloading a .splane file, which you then have to open with SimplePlanes (the same way you open any other file) for it to download the airplane you want.

Once it does it'll show up in the designer, you should probably also save it so you don't lose it when loading a different plane.