Next you have to press the strips of fabric in half lengthwise with the wrong sides together just as you do for bindings so that the right side of the fabric is on the outside.
Next you take your first strip and place it on the edge of one side of the quilt (raw edges to the edge of the quilt) and sew down using a 1/8" seam. Yes this seems rather small but there will be a second 1/4" seam going down when you sew on the large outer border so everything will be secure. The reason why I only use a 1/8" seam is that you don't want any stitching lines showing after the outer border goes on if your two 1/4" seams happen not to match exactly.
You continue the process with the other sides ensuring that border covers over the edge of the quilt. I personally find if I extend this inner border about a 1/4" beyond the quilt and start sewing it feeds better through my machine. This is just my preference.
Next you go and sew your regular large outer border on like you normally would. Here you see that my outer border fabric extends well beyond my quilt, this is because I'm doing a mytered corner border. Here you want to change you foot to a 1/4" seam foot which will let you guide the fabric for a consistent 1/4" seam.
This is the end result.