Meteors burn once they reach the mesosphere because that is the first part of the atmosphere with a non-negligible density of gas molecules. Although the air is still thin in the mesosphere, there is enough to cause friction and therefore heat for meteors passing through.

The higher layers of the atmosphere (thermosphere; most of the ionosphere; and the exosphere, which is nearly outer space) have almost nothing - not gas molecules or particles like dust - that could the meteor need to push through, and therefore nothing to cause the friction that heats up and then burns the meteor.

The reason why meteors usually burn up in the mesosphere is because the air in the mesosphere is dense enough that the meteor's moving through it creates a lot of heat (unlike the ionosphere), but the meteor doesn't survive long enough to reach the even denser stratosphere, let alone the denser yet troposphere. However, the larger and denser a meteor itself is, the longer it will last, and a meteor that is large or dense enough will make it into the stratosphere, the troposphere, or, if it is quite large or dense, it may even strike the ground.

Meteorites are meteors that hit the earth. Here's a link about a meteorite that was a fireball when it exploded close to earth. It was probably below the mesosphere when it exploded.

Here is a link about meteors burning:
link.