Possible causes preventing the construction of Units:
The loyalty of the region is too low or the region is in riot (i.e. a demonstration card has been played there)
I am sure that this is widely known among PBeMers who play Demo cards on each other all the time, but since Athena rarely uses them on the human player, I never noticed this effect before. Presumably it means you can't order new troops while a Demo card is in effect, but that troops that are already building continue to finish normally.
Papering a state's best build points with Demonstrations slows the ability of the enemy to build forces in the area, possibly allowing you to get the jump on them. St. Louis (before the Massacre) and Lexington MO would be useful places to play the card as the CSA, as would Cairo/Indianapolis/Evansville/Cincinnati to impair early Union plans in TN and KY. If you are attempting a Gold Rush, the Demos temporarily prevent the Union from building in CO increasing your Ranger's chances of successfully capturing Denver/Golden City early. Rather than use them on cities all over the map, keeping some in reserve to use back-to-back would prolong the effect on important build-points (although eventually the Union cities will calm down and unused Demos will be lost).