This is a great question, but the answer is a little complicated, so you have to help me out.

First, we have to understand air pressure. A lot of people think that air has no weight. This is not true. Air is matter and it has mass. “Mass” is a fancy way of saying “stuff.” We measure the mass by weighing it.

Pressure squishes things down so that they take up less space. “Volume” is a fancy way of saying “space.” The same amount of air is squashed into a smaller space down at sea level. On top of the mountain, it is more spread out.

Heat is energy, not stuff (matter). We can’t see heat. We measure it as temperature.

This part is tricky, so help me out by grabbing a piece of paper, a ruler, a pencil, and a red crayon or marker.

In your drawing, the red lines are to show heat. You can see that a lot less of the big square is covered by red. If you were an ant walking around in the big square, you would touch a lot less heat.

Heat is not really red, and it’s not really lines. The picture that you drew helps you think about heat. Your picture is a model. Scientists use models to help themselves think about things and to communicate to other people.

Some places on land are actually below sea level. Do you think they are hotter or cooler than places at sea level?

If you like to think about this sort of question, you may want to study atmospheric science.

So you know that the sun is really hot, but space is really cold. Since the temperature of the Earth is so much hotter than space, you’d expect that the closer you get to space the colder it gets. Eventually as you get closer to the sun, the temperature will start to increase. You may have heard people say that the air is “thinner” at higher altitudes. What this means is that it is less dense. Generally, the less dense air is, the colder it is. So the air is densest closest to the surface of the Earth, then it gradually becomes less dense until you get to the vacuum of space. This density difference accounts for the temperature difference.

At higher altitudes, the air is thinner. That's why it's harder to breathe at high altitudes--there's less oxygen. There's also less of everything else--carbon dioxide, water vapor, etc. These gases, which are greenhouse gases, are the part of our atmosphere which keep Earth at a warm temperature compared to outer space. Without this atmosphere, Earth would be uninhabitably cold. So, even though higher altitude areas are closer to the sun, they have less ability to absorb the warmth of the sun because they have less of these gases.

Good question!
When air expands, it has to push the surrounding air out of its way, which means that it expends part of its energy to do the pushing. As a result, the expanding air cools. When air contracts, it gets pushed into a smaller space by the air around it, which means that energy is put into it, which heats it up. Eventually, the expanding or contracting air will reach the same temperature and pressure as the air surrounding it, and the heating and cooling will stop. Air at higher altitude is under less pressure than air at lower altitude because there is less weight of air above it, so it expands (and cools), while air at lower altitude is under more pressure so it contracts (and heats up).

Air in our atmosphere moves up and down as part of the weather: the sun heats up the ground (which absorbs more light than air and is thus warmer than the air), and the air in contact with the ground heats up, and expands (and then cools). Elsewhere, cooler higher-altitude air sinks, is compressed as it descends, and gets heated as this occurs. This process is called "convection", and it is responsible for nearly all of our weather.

You are correct that it is colder at higher altitudes. However, the distance to the sun has nothing to do with this – it is the lower atmospheric pressure that results in lower temperatures. Keep in mind that the Sun is about 93 million miles (490 billion feet) away from Earth. Even if you hiked to the highest point in Santa Barbara (La Cumbre Peak: 4,000 ft) or were flying in a commercial airplane (cruising altitude ~ 30,000 ft), you are still relatively the same distance from the Sun (REALLY far away).

The answer to your question deals with the pressure and temperature relationship of a gas. The Earth's atmosphere is thin layer of gases composed of nitrogen (78%), oxygen (19%), and other gases (~3%). As you go to higher altitudes, there are less air molecules pushing down on you (lower pressure). When the pressure of a gas decreases, the temperature also decreases (the reverse is also true – when the gas pressure increases, the temperature increases). Therefore, the air temperature is lower at higher altitudes.