What kind of exterior wall insulation is this?
I'm not really seeing the part you're referring to with it being tucked in, but it looks to me like this is just foil-faced fiberglass batt insulation. The foil acts as a radiant energy barrier.
It is built like paper-faced insulation, but with a metallic facing instead of just paper. This is supposed to add some insulation value by reflecting energy back into the conditioned space or reflecting back to the outside, depending on how it's installed (you would face the foil inside if you mostly have to heat your interior, and outside if you mostly cool the interior).