How does heat even out in the real world?

Let there be a room that can be considered an isolated system at a certain temperature, say 25°C. The outside is a heat reservoir of 10°C.

I know that due to the laws of thermodynamics, if I allow the systems to come into contact and allow thermal transmission, the room's heat will transfer to the outside until it is at 10°C too.

However, I don't understand how it actually happens. Once I open a window, for example, how does the air in the opposite side of the room know that a window has been opened, and move its heat outside?

Maybe it helps you to visualize it on a molecular level. Temperature is just specifying the average kinetic energy of the molecules. That means inside your room at 25°C the molecules have on average a higher velocity than the molecules outside at 10°C. If you open the window some faster molecules fly through the window to the outside and some slower molecules fly to the inside, thus the temperature in your room decreases. If the inside and outside temperatures are the same, there are for a given kinetic energy (or speed) on average as many molecules travelling from inside to outside as the other way around. Therefore the temperature stops changing.

Even if you you leave the window closed, air molecules will hit the inside of the window and lose energy due to collisions with the vibrating glass molecules. That glass molecule will make other molecules further in the glass vibrate too and so on, until the glass molecules on the outside vibrate. If a slower air molecule from the outside hits the hot glass it can get a push and move faster.

how does the air in the opposite side of the room know that a window has been opened...

The flow of heat arises purely from random events (random based on our limited knowledge of the position and velocity of every single gas molecule). The air can't be aware of anything. The high energy particles inside will move out and collide with the lower energy particles, and vice versa (this happens by chance, but the chance is very very very likely). As these molecules mix, their energies average out until net equilibrium is obtained.

the room's heat will transfer...

(and)

...and move its heat outside?

Another misconception to tackle: the air doesn't have heat. It has energy. When energy is transferred by purely random mechanisms, we then call this energy transfer heat. But the warmer air doesn't have some stored amount of heat that it can release. The correct word to use is energy.