How do I apply foam board insulation behind a new recessed shelf in an exterior wall?
First of all, recessed shelving in exterior walls is not a great idea. You won't actually gain much space and the shelves will infringe on the space available to place insulation, run electrical lines, etc. I'd prefer doing this in an interior wall, or only on an exterior wall in a house with much thicker walls.

But if you want to proceed, here's how.

The black fabric sheet is probably tar paper. Your house is probably one of many in the bay area that for some bizarre reason was built with no structural sheathing and instead, tar paper and wood lath+stucco right over the studs. To insulate such walls without causing many terrible problems requires special attention to detail. Don't put in any dedicated vapor barriers. Almost no houses actually need them and they usually cause far more problems than they solve.

In your case, rigid foam boards are actually a reasonably good choice, provided you always follow these guidelines:


Leave a 1/2" to 3/4" gap between the foam and the tar paper--very important! This is an air gap to promote drying.
install the foam board such that the foil side is facing the tar paper. The foil side is a radiant barrier, and when it faces an air gap, it works and gives you a little better performance, especially in hot weather.
Scrupulously caulk or spray foam the edges of the foam board to the studs.


The foam board you have chosen is EPS--expanded polystyrene. It has about R-3.8 per inch. So if the boards are one inch thick and you install no other insulation, you will be adding about R-3.8 to the sections of the walls that are between the studs. If you did this for your whole house, assuming that 25% of the wall is actually studs @ R-1.2/inch, that would give your whole wall about R-3.9 (1.2*3.5*0.25)+(3.8*1*0.75), which is still really low. I would advocate using thicker foam or filling the remainder of the cavity between the foam and the new drywall with more insulation. Fiberglass or mineral wool batts or simply more foam would all be DIY options.