To add strength to the shelf and cover up the edges of the plywood, we'll be using 1x2 (on edge). This 1x2 will have 45-degree miters in the corners and will be affixed to the plywood shelf using biscuits and glue.
To begin, make a 45-degree miter cut on a piece of 1x2 (placed on edge on your miter saw). Then, measure the width of the plywood (15-1/4" in my case), and square cut the 1x2 at this distance, measuring from the short point of the miter cut.
Next, miter cut a second piece of 1x2 a couple of inches longer then the length of the face of the shelf. Position this piece on the front of the shelf, and align the previously cut piece on the side, making certain that the miters match up nicely in the corner. When satisfied with the position, set the front piece of 1x2 aside and make a couple of light pencil marks across the plywood and 1x2 at positions where you want to place a biscuit. Two or three biscuits should be appropriate for the side of the shelf. Also, mark the final length of this piece (at the back side of the shelf) and square cut the piece with your miter saw.
Now, after setting up your plate joiner for a 90-degree cut at 3/8" depth and preset for a #10 biscuit, make the biscuit slots at the points you marked on the edge of the side of the plywood shelf, as well as at the points you marked on the 1x2 for the side of the shelf. Place a small, even amount of glue into the slots on the shelf and insert a #10 biscuit into each slot. Then, place glue into the slots on the side piece of 1x2 and attach it to the biscuits on the side of the shelf. Re-position the front piece of 1x2 to check the miter joint and adjust the side piece (by sliding it fore or aft on the biscuits) if necessary.
You may wish to use a couple of finish nails or clamps to hold the side piece of 1x2 in place while the glue dries.