The thermal system around a modern processor chip is indeed complicated and a major design focus. For both electrical and economic reasons, it's good to make individual transistors in a processor small and close together. However, the heat comes from these transistors. Some is dissipated all the time just because they sit there with power applied. Another component occurs only when they switch states. These two can be traded off to some extent when the processor is designed.
Each transistor doesn't dissipate much power, but millions and millions (literally) crammed together in a small area do. Modern processors would cook themselves in seconds to 10s of seconds if this heat wasn't actively and aggressively removed. 50-100 W is not out of line for a modern processor. Now consider that most soldering irons run from less than that, and heat a chunk of metal with about the same surface area.
The solution used to be to clamp a big heat sink onto the small die. In fact, the heat sink was a integral part of the overall design of the processor. The package has to be able to conduct the heat power from the die to the outside, where the clamped-on heat sink can conduct it further and eventually dissipate it to flowing air.
This is no longer good enough as the power density of these processors has gone higher. High end processors now either contain some active cooling or a phase change system that moves heat from the die to radiating fins more efficiently than plain old conduction through aluminum or copper did with the old heat sinks.
In some cases Peltier coolers are employed. These actively pump heat from the die to someplace else where it is easier to couple to the air flow. This comes with its own set of problems. Peltiers are rather inefficient coolers, so the total power that needs to be gotten rid of is significantly larger than just what the die dissipates. However, the active pumping action can help, even if the radiating fins eventually are much hotter. This works because the aluminum or copper of the radiating fins can stand much higher temperatures than the semiconductor die can. Silicon stops acting like a semiconductor at around 150°C, and real circuits need some operating margin below that. However, heat sink fins can easily handle much higher temperatures. A active heat pump makes use of this difference.
In the past there have been processors cooled with flowing liquid nitrogen. This doesn't make economic sense for ordinary desktop PCs with today's technology, but heat management has been a important part of computer design pretty much since the beginnings of computers. Even back in the 1950s, keeping all those vacuum tubes from melting each other was something that had to be carefully considered.