What benefits did the 3 field system have on the population?

Resting ("fallowing") a different third of the land each year in theory allowed a third more food to be grown (as two-thirds would now be cropped rather than a half) and allowed for the introduction of legumes which returned nitrogen lost by the soil when planted with cereals, though fertility still relied heavily on the availability of livestock manure (perhaps the reason why many areas retained the two-field rotation, which might be less demanding of additional soil nutrients). So (again in theory) a third more people might be fed, though this would vary according to local conditions.