What is the purpose / function of media converters on Single-Fiber mode cable runs? How does this work or is necessary?
There is nothing special about media converters in regard to single-mode fiber. A media converter would typically be used to extend the reach of copper (using fiber in between the media converters) or to accommodate endpoints that don't support the media in use. For example, if you have fiber between two endpoints that only support RJ45 copper media. You could use media converters to convert the fiber/optical to copper/electrical.
Your question is vague, to say the least.
I run a small campus single-mode fiber optic network without any "media converters" (so-called.) I do have switches with both copper and SFP ports, and those do convert from one transport to the other, generally at far more attractive prices and with better performance than most "media converters."
Whether done by a "converter" or a switch, conversion allows the data to flow to devices with different interface types, or through a medium that has better performance (such as length between nodes) that's between two devices with the same interface type. I can't run 4000 feet of copper between two computers, but I can run 100 feet of copper to a switch connected to 3850 feet of fiber connected to another switch connected to 50 feet of copper connected to the other computer; And when a thunderstorm comes along, I'm very happy not to have copper between them.