Heat transfer happens by heat conduction and heat radiation, so even in a vacuum you will radiate heat away in the form of infrared photons, at least if the vacuum is not at a higher temperature. Heat radiation is however a less efficient process then heat transfer in a sufficiently dense environment. That is also an issue for spacecrafts, they need rather large radiators to radiate away excess heat into space.

You lose/gain heat through conduction and radiation. Eliminating/reducing the atmosphere or making it out of something less conductive or lower specific heat reduces the speed at which the vibrating molecules of your body transfer kinetic energy to or gain energy from the molecules of the surrounding environment.

The vast majority of heating/cooling that people experience in their day to day lives is done through conduction from the earth's atmosphere so in most situations you'd lose heat more slowly on Mars even if it is colder. It's kind of like the difference between being in cold air and cold water.

You lose out on convection and your heat loss has to be almost entirely through black body radiation instead. Astronauts have to work harder to stay cool than to stay warm, especially if the sun is shining on them.

I think that property of a vacuum is already used in a day-to-day object.

https://en.wikipedia.org/wiki/Vacuum_flask

Vacuums make very good insulators, that's how a thermos works. Any cool gas is cooling you down but the denser it is the faster you cool down.

One of the major sources of headaches for designing satellites is thermal management.

In fact, during some weeks of the year the Space Shuttle could not visit the International Space Station because at the ISS inclination, the Shuttle would have spent too much time in the sun. The ISS, Space Shuttle, and other satellites have large radiators to cool them down.

That is correct, things don't cool down easily in a vacuum. Overheating is a problem on space vehicles. Thermal drinking bottles use a vacuum layer to keep heat in.

> help me wrap my head around that

Think about when it's cold, and when it's cold and windy.

Low pressure air is like "anti" wind, even less cold then no wind on earth.