How is the slave economy of the Slaver’s Bay cities viable? Why wouldn’t the Free Cities just breed their own slaves instead of buying them elsewhere?

As with anything, look first to division of labor and specialization as driving factors. The cities of Slaver’s Bay produce skilled labor – the Astapori have a monopoly on Unsullied, the Yunkish produce sex slaves, and the Meereenese specialize in pit fighters. If you want any of those three, then rather than having to go through the expense and difficulty of setting up training operations, you just send an order. 

Meanwhile, while the cities of Slaver’s Bay focus on turning the Dothraki’s “raw materials” into “finished goods,” the rest of Essos can concentrate on finance, manufacturing, commerce, and agriculture. 

It’s a vile, immoral system but it works economically.