Ooh hmm. It looks like if you leave the top part of the seam un-sewn. You'd need two extra pieces (probably triangles) of fabric tacked over the open part of the gusset and sewn in around the edges before adding the lining.
OR you could make the outer part of the bag a bit wider and then bring the edges together at the sides, creating a gusset/pleat by doubling up the fabric.
So you'd make the two outer fabric pieces X" long (however long you wanted the purse to be, plus four inches or so), sew the two sides of the bag together (the short sides), right sides together.
Turn right side out.
Measure out an inch or two from each seam and then mark with an erasable pen.
Pull those two marks until they meet, making a pleat - pin.
Turn right side in and pin up the side until it's where you want the open gusset to begin and sew that seam. I'd probably backstitch at the top, too, to give it extra reinforcement.
Press the original side seam so it lies flat against the seam that makes the pleat.
Turn right side out. It SHOULD give you a nice side gusset pleat. Might be a bit thick, with the interfacing and all, but it wouldn't be unmanageable.
Not sure if these directions make sense to anyone who isn't me, though. But you've got me thinking and now I want to make a bag like this. I may have to try that. I've got some awesome fabric for it, too.
if I do, I'll take in-progress pictures.ETA:
Or you could make the pleat the easier way and ...
Sew up both sides.
then move in an inch (or so) and sew up 3/4 of the way up on both sides (or more, depending on your preferences).
Turn right side out - you'll have a pleat (it'll require pressing, though.)