Picture peeling bricks off the outer wall of a triangular brick pile, one brick at a time, always taking from the exposed boundary so the remaining pile never falls apart. Some bricks touch the outside along a full edge and some only at a corner, but either way, removing one always trims the same fixed amount off , and together, leaving untouched.
Working inward from the outer face, remove triangles one at a time. A triangle that touches the outer face along a single edge is deleted by erasing that shared edge, merging it into the outer face: this removes one edge and one face and leaves unchanged. A triangle that touches the outer face at a single vertex, with its other two edges on the boundary, is deleted by erasing those two edges together with the vertex opposite the outer face: this removes one vertex, two edges, and one face. In both cases does not change, and the process always leaves behind a smaller, still-connected triangulated graph.