Once upon a time knit borders were always made separately, then sewn onto the shawl, sleeve, hankie, hem, or neckline later. These days we’re all about minimizing time spent sewing one knit thing to another, and avoiding sewing whenever possible. The term “applied border” is just a fancy way of saying you’re going to knit a border and attach it to the main body of knitting as you go.
What this means is that, in some cases, every right side row ends by attaching the border to the body of the shawl with a k2tog (or ssk), where the first stitch on the needle belongs to the border that you’re knitting now, and the second stitch of the k2tog (or ssk) belongs to the body of the shawl, sleeve, hankie, hem, or neckline. As the knitting progresses, the total number of live stitches along the shawl edge decreases, eliminating the need to bind them off. Other times the join is made on wrong side rows (usually with a p2tog), and sometimes you’re picking up a stitch along an edge to work the join, instead of using stitches that are already on the needle.