Why does producing a single bacterium through genetic recombination enable researchers to produce large numbers of bacteria with the recombined...

Because bacteria reproduce! They are alive!

It's difficult to overstate how momentous an achievement it is that in recent years we have been able to cross that threshold into creating synthetic lifeforms. Very simple lifeforms, mind you; all bacteria. But life. We have crossed a threshold in the evolution of our civilization that is nothing less than Promethean---it belongs on a list with fire, the wheel, the steam engine, the atom bomb. Synthetic biology will open previously unimaginable applications, with tires that self-heal so they never go flat and bandages that grow directly into your skin to become incorporated as permanent tissue being some of the most mundane. More radical applications include terraforming Mars and living computers.

Bacteria not only reproduce, they reproduce extraordinarily fast. The famous E. coli can divide itself about once every 20 minutes, which creates exponential growth that is only constrained by the available nutrient substrate (usually agar in a petri dish).

So, if you start with one synthetic bacterium, in 20 minutes you have 2. In 40 minutes you have 4. In an hour you have 8. Not so impressive, right?

Ah, but wait. In 2 hours you have 64. In 3 hours, 512. In 4 hours, 4,096. By the end of an 8-hour workday you have over 16 million, and I guarantee you that when you come back in the morning your petri dish will be filled, no matter how big it is, because if the exponential growth had been allowed to continue then within 24 hours the bacteria would have engulfed the Earth.

Thus, creating that one bacterium might be more difficult than even making thousands of the plasmid we want by more direct means---but once it reproduces, the bacterium won't give us thousands---it will give us millions, billions, or even trillions.