Meteors burn up in the mesosphere when they encounter the molecules from that layer and rub up against them. This generates heat; ironically, the mesosphere is the coldest part of the atmosphere.
When a meteor begins to enter the mesosphere, it immediately bumps into molecules of the Earth's coldest atmosphere and scrapes against them, causing friction and rapidly generating heat due to the speed of the meteor. The meteor begins to glow, and chips of the rock start to fly off. Although the air in the mesosphere is cold and thin, the presence of gas particles causes objects to heat up as they move past through the air.