The machine sews any flat parts of the design first, then the operator lays a sheet of embroidery foam over the area. Satin stitches are sewn across it, and the needle perforates the foam along the edges so the excess can be torn away. A little heat is sometimes used to shrink stray bits that peek out. Foam comes in colors so it can match the thread.
Puff needs its own kind of design: bold shapes, thick letters and satin columns wide enough to wrap the foam cleanly. Fine detail, small text and large fill areas do not work in puff, though they can sit alongside flat stitching in the same logo. Structured cap fronts are the classic home for it.
Ask whether your logo can be digitized for puff as drawn or needs simplifying, and request a sew-out. It is a different effect from puff ink; see the 3D puff embroidery guide and puff on hats vs garments.