There is a round copper insert attached to linkage that slides up and down inside the overflow tube which operates with the lever. Where the tub drain connects to the overflow there is a tee. This insert drops down into the middle of the tee and blocks the portion of the tee where the tub drains stopping the water. When you want it to drain, moving the lever moves the insert up and out of the way.
At the hinge point for the lever, there is a spring behind it to help hold it in position. The spring could be worn as well as the hinge area itself. If so replace the spring or the whole plate. The insert and tube can clog up with hair, etc. that could affect whether it holds but I just replaced one a while back where the spring and hinge area were worn to bad (good luck just trying to find a spring). Make sure if you pull it out that the adjusting nut on the linkage is tight and if you adjust the length at all then you may have to play around adjusting the linkage to get it to hold and drain.
I kind of like the step-toe drains so you don't have to mess with that except if you don't watch it, the step-toe turns into a stub-toe.