As Alexey pointed out, you have to use either the Route or the Closest Facility solver to get the actual paths.
In you case, the "flow" is most likely on the demand points. So after solving the location-allocation problem, the demand points has the Facility ID each demand point is allocated to. Now when you load these locations into a Route or CF layer, you can "map" extra fields. During the load location dialog, there is a "Location Analysis Properties" section with titles of Property, Field and Default value.
You can ADD new fields on the Stops attribute table or the Incidents attribute table and they will show up in the property list and you can map the flow value over.
OR you can do a join of that value after the fact.
Also take a look at the GP tool "Copy traversed source features" in 10.1 that you can run after solving the Route or CF to generate a line feature class with every edge that was traversed by these solvers (if needed).