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Think of the atmosphere as a layer cake. The bottom of the cake is very dense and the higher up you go, these layers get less and less dense. Up on a mountain, the air is very thin.
In other words, there are fewer air molecules per cubic foot (volume of air). The molecules are farther apart and can hold less heat energy. Because "heat" is what we say when we mean molecules are moving around. The more they move, and the more molecules there are, the more heat you have.
It's also helpful to know that the air is heated by the ground and oceans. The heat from our sun mostly passes right through the atmosphere, not directly warming the air up very much. But the surface of the planet will warm up wherever the sun is shining on it. And in turn, the warm ground or the warm surface water then gradually warms the air from the bottom up. (This is because heat is transferred in different modes: radiant, convection, and conduction.)
And the warm air does indeed rise. As it rises, it gradually spreads out and cools off again. Some of the heat even radiates back out into space.
There are "fountains" of air constantly circulating throughout the atmosphere, and this creates weather patterns. It tends to snow on mountains because the warm air has carried some moisture with it on its way up. As it cools and thins, it can't carry the moisture any more, and the moisture precipitates out. Which is why we call it precipitation whenever it snows, rains, sleets, etc.
So by the time air reaches a high mountaintop, it's probably going to be cool or even frigid cold.
This is also why hotter regions, like the southern US, tend to get very humid in the summer. The warm air can carry a lot of moisture, and there is a lot of warm surface water. Our sweat is less efficient when the air is moist, because it takes longer to evaporate and carry the heat away with it.
Deserts have few water sources. So they also have hot dry air, and much less humidity, and therefore little to no precipitation. But they also get cold at night, because there's very little humidity to hold the heat overnight.
All of this is to illustrate the complex interactions between the sun, the atmosphere, and water (or lack of it) on the surface, and humidity in the air.
Inside an older building you're more likely to experience warmer air on higher floors than lower floors because the air is trapped in a nearly closed system and hot air rises. Of course, HVAC engineers try to compensate for this in modern buildings.
While I agree in general, one point is a bit to simplified in my opinion
Less molecules mean less heat, it has nothing to do with the temperature, if you just decrease the density by removing half the molecules, you have the same temperature.
It cools down because it expands adiabatically. Consider a very thin balloon filled with air which is warmer than the surrounding. This now rises up, but as it does, the pressure decreases, causing the balloon to expand. During this expansion, the balloon transfers energy away from itself, because it has to push away air, to make room for expanding in the surrounding. This work cools the air inside the balloon. Assuming the air inside is dry, it would cool around 10 °C per km it rises. Now if you think about it, the balloon just stopped the inside from mixing with the outside. If you look at a large "piece" of air, it does not mix very fast, so you can remove the balloon and just consider what happens with warm air heated from the ground.
Now this does not mean, it has to be cooler when higher up. The same points hold, inside a house, but there it is often warmer when higher.
The best explaination is when looking where the heat comes from and goes too from the air. The atmosphere is mostly heated from the surface of earth, so the bottom and cooled from the upper layers. So naturally it gets hotter where it is heated. The question is now by how much? There are three modes of heat transfer in the atmosphere: radiation, conduction and convection. The first two are very slow. Connection is fast but has limits. Consider the piece of air, if it rises, it cools. So at some place it may be the same temperature as the surrounding air, so it stops rising. This means the convection works only when the air gets cooler by 10 °C/km going up (~6.5°C when the air is moist and precipation happens). So this temperature gradient is observable very often.
But does the energy to expand the balloon not just come from the pressure? Pressure has units of energy per volume btw.
Balloons are open. Most typically do not expand but the excess air just escapes out the bottom. Basically they will rise till the overall weight matches that if what they displace.
There are more efficient balloons that do expand and can attain same great heights. Far more than conventional aircraft even. But that expansion is mostly due to excess material in the construction and little from stretching. Thus the pressure difference is minimal while the volume increase significantly with altitude.
I can't even criticize this comment, because you wouldn't understand why.
So thanks for the effort I guess? But we're not discussing real balloons.