Heat can transfer through conduction (basically thermal diffusion through physical contact), convection (bulk motion of matter, like gas or water flow), and radiation. For a spacecraft in low Earth orbit, the pressure is considered ultra-high vacuum, so you basically only have radiation to dissipate heat. Near room temperature, this would be mid-infrared light. The energy in everyday sound waves is very small, so body heat, on-board instruments, sunlight, and perhaps even IR emission from the Earth would be much more important contributors to heat build-up. However, regardless of the heat sources involved, there will always be some equilibrium temperature where the energy going into the system equals the energy radiating away.

To keep things comfortable for the crew on the ISS, there are passive and active systems to regulate the temperature [1]. For dissipating excess heat, large radiators are used. These are basically panels with a large surface area in order to maximize emission of thermal radiation. A closed-loop system is used to circulate fluid, which collects and transfers heat to these radiators. Water is used for some parts, but others have pipes on the outside that use ammonia to prevent freezing. The radiators themselves can be retracted or deployed as needed.

[1] Memi, E. G. "Active Thermal Control System (ATCS) Overview." (2006): 19.

I don't have a great answer for you why, but heat must be radiated away from space ships faster than you might think. They have heaters on them to keep them warm. Think Apollo 13 when they turn off all their power, and it gets cold.

The ISS is traveling through a decent amount of atmosphere still, hence they need to boost their orbit occasionally. That atmosphere is probably plenty to dissipate WAY more heat than sound creates.

That doesn't explain deep space ships... But they do clearly radiate heat, if not slowly. But probably faster than what little heat sound creates. (Also think of the cooling phase that James Webb space telescope went through)