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I've looked at several sources now, and still haven't got a clear mental picture of exactly how that works. I suppose it must, since the aircraft don't usually fall out of the sky, but it's hard to see a reason why such complexity would be necessary.

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Tabs are a way to reduce the amount of stick force (i.e. force applied by the pilot) required to deflect a control surface in a mechanically linked system. In most modern aircraft hydraulic or electric systems fill the same purpose

To give a simplified explanation, a small tab located at the trailing edge of the surface is deflected by the control linkage. The deflection of the tab generates an aerodynamic force which then creates a moment around the control surface hinge, moving the entire surface.

So if you deflect the tab upwards, that air flow over the tab generates a downwards force that acts to move the control surface down.

And another layer of simplification: the pilot controls small control tabs which fly the main control surface into the position the pilot wants it to be in, which then moves the wing.

Usually the connection between pilot intent and wing movement is:

Pilot moves the stick->Aileron moves, changing lift on the wing->wing moves

In the MD-80:

Pilot moves the stick->Tab moves, changing lift on the aileron->aileron moves, changing lift on the wing->wing moves

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