When something burns, the energy that is released by burning is released as electromagnetic radiation – most of this is released as infra-red radiation (heat) but some of it is emitted in the visible part of the electromagnetic spectrum.
The actual photons of light emitted are caused by a process called atomic excitation – the large amount of energy in the flame excites electrons in atoms and they are ‘promoted’ to a higher energy shell/orbit around the nucleus – but this is unstable, and so the electron released the extra energy as a photon and the electron falls back to its normal orbit/shell.
The wavelength of the photon depends on what is being burnt.
As Dave has stated, the heat in the chemical reaction ‘excites’ electrons to higher energy levels. When the electron drops back down the energy is released and has a characteristic wavelength based on the type of atom being burned. This wavelength gives rise to the light we see and the colour differs depending on the wavelength.
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