Radiation is the part you are missing. The three ways of dissipating heat are: conduction, convection, and radiation which is what the "fins" do.

No. In the vacuum of space there is no convection. The only maintenance free(ish) method of discharging waste heat is to radiate it as infrared, which is not terribly effective compared to terrestrial heat management systems where we have the benefit of a big old atmosphere to dump heat into.

Radiative cooling into space is seriously weaksauce. The amount of heat an object can dissipate in such a manner is described by the Stefan-Boltzmann Law. It would take nearly a square meter (0.84 m^2^ according to my admittedly possibly shaky math) of perfectly ideal thermally conductive black body radiator material to dissipate the 640 watts of waste heat from just one datacenter style GPU at 70° C.

A square meter of heatsink. For one GPU.

Your radiator heat sink can't be shaped like a terrestrial one, either, with stacked fins providing a high surface area in a small volume. That's because a black body radiator is not only an ideal emitter of heat into a vacuum, such as it is, but also an ideal receiver. Your heat sinks will have to be wide and flat so they don't radiate most of their heat right back into other parts of themselves, and this also precludes putting your equipment near other pieces of equipment so they don't radiate their heat into each other.

A single server rack in an AI data center will consume and thus have to dissipate something like 80 killowatts, i.e. 80,000 watts, which even if you had access to some type of physics-experiment-land totally ideal radiator material with an emissivity of exactly 1 would require a 102 square meter radiator just to dissipate that same 70° C. And no part of it could be baking in the sun, nor be influenced thermally by any adjacent servers. In reality it'd have to be even larger, because such a perfectly ideal material does not exist.

TL;DR: Getting rid of heat in space is extremely difficult and in fact is one of the biggest challenges of spacecraft design. Thus putting massive heat generating computers in space is a self-evidently moronic idea as cooling them would be effectively impossible.

The short version is - if you've ever heard of infrared cameras that can see heat, that's because everything glows based on heat. When things are very hot the glow is visible, but even cool things emit light outside of the visible spectrum. And emitting light takes energy, which means blackbody radiation (as this is called) takes energy (heat) away.

Jesus Christ, how do you think the Sun works? How are people this clustering ignorant!!??