The lit match heats the air inside the bottle. As the heated air expands, some of it escapes out of the bottle. The greater pressure outside the bottle forces the egg into the bottle. Most solids and liquids expand when they are heated and contract when they are cooled. Gases also expand when they are heated at a constant pressure.
Heat causes expansion because it increases the vibrations of a material's atoms or molecules.
In a gas, heat also increases the speed at which the atoms or molecules move about. The increased movement forces the atoms or molecules farther apart and the body becomes larger.