Heat? A problem in space? Really?
Yes, really. I expected you to have been in enough of this type of discussion to know this already.
It is a common myth that the vacuum of space is cool. But it actually can be better described as lacking any temperature at all, as it lacks what could have a temperature.
So, umm, how does that big ole star warm us up without an air conduit between us? Or are you one of those 'old testament' types?
As I mentioned, you can try to radiate heat (well, it's not trying. You will do that). But it is a far slower process. The sun is so incredibly big and hot that radiation is enough to heat up Earth.
But, you may notice, the sun isn't exactly getting cooler while doing so either.
Temperature is movement of small particles (e.g. molecules, atoms). This type of movement generates radiation, which takes energy from the movement and reduces the temperature. But a far more effective way to cool a bunch of particles moving is to bring them in contact with a bunch of particles moving slower (e.g. being cooler). The particles will transfer their movement energy and the slower particles with get faster, the faster get slower, and the temperature drops (for the hotter particles).
The vacuum of space lacks all these other particles to collide with and transfer energy with. So all that is left is radiating heat. And that's a slow process. If your satellite or space ship is producing too much heat, it can't cool it down that way, and it will get hotter and hotter until it stops functioning.