The Expand tool is for removing fringes around the edges of objects when you make dramatic changes. Like the Smart Brush, it will expand which ever area you hover the center of the brush over. However, instead of selecting large areas, it just expands the selection by a single pixel inside the brush. You can use the brush multiple times to continue expanding the area.
Tips: The Expand tool makes very subtle changes. Zoom in and out using your mouse wheel, the + and – keys, or the Zoom slider in the bottom right of the image.