The underlying computer models built into the magic black boxes will know the relevant parameters which determine lift from onboard measuring systems - incidence (AoA), density, speed, and a reference area to balance up the units. In addition (and most folks either forget or are not aware of these) Reynolds Number and Mach Number affect the shape of the lift curve slope (incidence against lift coefficient) as well as incidence.
If we take the simple case of cruise flight, for instance, we can put
lift = weight = some constant x lift coefficent x speed squared
so, if we know the weight and the speed, then we can figure out what lift coefficent we should have for a given set of conditions. The result is that we know at what incidence the aeroplance should be flying for that lift coefficient.
If the calculated incidence is higher or lower than the real world incidence on the day, then the presumed weight is wrong. It follows that we can reverse the sums to figure out what the weight is.
So it seems that the the weight is calculated to display speed tape information. I think I'll go re-examine the report on the Emirates A340 incident in Melbourne. I wonder what the speed tape looked like in that instance.