To answer this question, we need to study the Lift Equation, the principle by which planes can fly.
For an aeroplane to fly, the lift it produces has to be greater or equal to its weight. Assuming the speed, size and the shape and position of the wings remain constant the only value which is independent of the aircraft is the density of the atmosphere. Greater density provides greater lift.
This is the reason that aircraft can fly easily at lower altitudes but can only fly up to a certain height (called a ceiling) due to the reduction in the density of the atmosphere.