WHAT
Stay stitching is an important preliminary step to constructing your garment. A stay stitch is stitching along a raw edge of a single pattern piece in order to prevent stretching or neckline or armhole.
WHY
It ins't uncommon to skip this step when you are more experienced with a particular textile or garment, but it is definitely a good idea to stay stitch when you are constructing something for the first time, or are deciding to design/self draft your own pieces. This will allow you to critique your pattern without external factors such as fabric stretching affecting your future changes you may make to your pattern.
There are a couple reasons why you want to stay stitch in terms of fabric behavior. Typically this step is done to a curved edge of an arm hole or neckline. Curved edges of a garment will be cut on the bias, which means that edge will exhibit the characteristic of easily stretching. You want to counteract that characteristic so that your garment doesn't warp/stretch as you are constructing the garment (there is a lot of manipulation to the fabric when constructing a garment, I have had pieces stretch over an inch along a neckline, which is very problematic when applying a collar or collar stand).
HOW
Stay stitch within the allotted seam allowance of your pattern. This stay stitch is stitched 1/8 inch from edge.
For crew necklines I prefer to start on center front, this not only marks the center front point, but was told once in one of my first sewing classes that this is the proper way to do it so the results of slight stretching are symmetrical.
Beginner tip: If you are worried about stitching at 1/8 from edge, stitch 1/4 inch from edge with a longer stitch length that can be easily removed with a seam ripper if you see it sticking out of your neckline finish. I find the length you choose for a stay stitch is a personal preference, you may keep it at a normal 2.5mm stitch length, or make longer ( about 3.5mm) for easy removal. The down side of a longer stitch length is that it will be harder to stitch along an edge as you will move faster along a curve.