This one is prime example of an argument based on ignorance.
First off, you need to understand what air pressure is. Air is a mixture of molecules which are constantly in motion, bouncing off of each other and anything else in their way ( such as the wall of a container). Air pressure is a measure of the force per sq unit of area these molecule exert on a surface they are colliding with. A vacuum is just an absence of said molecules. If we have two chambers, side by side, one with air in it, and one a vacuum, with a divider between, you will have air molecules bouncing off one side of the divider, and none on the other, and you have a net force acting on the divider. If you were to remove the divider, molecules that would have hit the divider are now free to travel into the other chamber. Essentially, the molecules are given more room to spread out and they do so.
Now put the vacuum chamber on top of the air chamber, and remove the divider again. The air molecules will again start to spread out, but now, in order to move into the upper chamber, they have to climb. They do so at the cost of losing speed. (just like when you toss a ball in the air; it loses speed as it climbs). Now the distribution of speeds of the molecule are not even and you are going to have faster molecules and slower ones. The faster ones will make it all the way to the top before bouncing off it, but the slowest ones won't. They'll slow to a stop and fall back to the lower chamber before reaching the top.
Put another chamber on top of the second one, and remove the divider. Again, some of the molecules that reached the top of the second chamber will still have enough speed to reach the top of the third chamber, and others will not. Keep adding chambers on top of each other. With each successive chamber, fewer and fewer molecules will have initially been moving fast enough to make it that high. Eventually, you'll reach a point where no molecules ever reach the top of the chamber, and even if the top chamber was open to a vacuum, no air will escape.* You will end up with higher air pressure at the bottom of the column of chambers, with decreasing air pressure as you move upwards.
This is what happens on the Earth; air molecules attempt to spread upwards, but don't have enough speed to get too far from the surface before they "fall back" At greater and greater heights, fewer and fewer molecules make it that far, and the air gets thinner with altitude. ( as noted in an earlier post).
* Though on occasion, a rare molecule will, by a chance series of collisions, pick up enough speed to escape the chamber, But this type of "leakage" would happen at an extremely slow rate. For something like a planet, it would take billions of years to make a significant difference.
