A queen can lay about 2000 eggs per day. For brief periods of time, she may be able to lay 4000 eggs per day, but most queens can't sustain this rate very long. This means that once the colony stabilizes, at least 2000 bees are hatching and 2000 bees are dying daily. Since most adult bees in summer live about 36 days, the maximum population of the colony will peak somewhere under 72,000 bees. Multiply 2000*36 to get this number. Now think about this a bit and figure out what happens if bees live up to 7 days longer. Now we have 43 days times 2000 bees hatching daily which gives a peak population of 84,000 bees.
Think of bees that just emerged as "fat" bees with lots of reserves. Feeding brood consumes these reserves depleting a limited resource thereby reducing the life span of each bee. This is where short lived "summer" bees come from. Long lived "winter" bees expend much less energy feeding brood and can live 5 or 6 months. Reducing the time a bee spends feeding brood increases that bee's lifespan so it can forage longer.
With 20,000 bees in a cluster, almost all of them will be required in the hive tending brood. Bees that forage under this paradigm will mostly collect pollen. In early spring, it is typical for 80% of the foragers to be pollen collectors or to collect both nectar and pollen if they can. The net effect is to severely limit the amount of nectar a colony can collect from early spring flows such as maple and fruit bloom and to reduce the lifespan of most workers in the colony because they are feeding brood.
By placing two queens and therefore two clusters of bees into a single hive, they share heat so that about 10,000 of the 20,000 bees in each cluster are now able to forage. Foraging bees don't consume as much of their stored energy as nurse bees therefore they live longer and they start foraging a few days earlier than if they had been needed as nurse bees. Extending the life of bees in a colony in spring by just one day means 2000 more bees in the colony. In addition, the pollen needs of the colony can now be met by a much smaller proportion of the foragers. This means more of the early spring foragers will collect nectar which can turn otherwise underutilized nectar flows into major honey producers.
There is one more effect that comes into play with two colonies. If they can share workers through a common space such as through an excluder, nurse bees will move to where they are needed most. This means the strongest laying queen will have abundant nurses even if her colony started out with fewer bees. This is why most two queen systems are set up so workers can freely move from one cluster to the other through an excluder.
Put it all together and a two queen colony uses less physical hive components, has longer lived bees foraging for nectar, can take advantage of early spring flows, and does this while balancing the nurse bee population according to need.
