To remove the background:

In the toolbar, choose Extract Shape from the Select pop-up menu.

Trace the edge of the image you want to save.
Make sure the image’s edge is within the thick border that appears as you trace.

Connect the end of the border back to its beginning, or double-click anywhere to extend the border in a straight line back to the beginning.

Use the resize handles to make sure the border contains the image’s edge, and then press Return.
Preview analyzes the border you drew, determines the outline of the image you want to extract, and washes out everything outside that outline.

If you can still see the background through a “hole” in the image, drag in the hole.
As you drag, Preview removes the dragged area and any pixels adjacent to the dragged area that have the same colors.

If you want to restore part of what you removed, Option-drag in that area.
As you Option-drag, Preview adds the dragged area and any pixels adjacent to the dragged area that have the same colors.

Press Return.