Step 1 - Understanding the interaction of electrons and photons
Bohr's model of the atom states that the electrons are orbitating the nucleus in shells that are increasingly distant from the nucleus:
Light can be understood as being composed of small particles called photons. When photons collide with electrons, i.e., when we throw some light in a sample of a substance, the electron of a level can jump up to the next level. This is called light absorption.
When the electron reaches an upper level, it will not be there for a long time, since the atom becomes unstable. It will come back eventually. When it comes back, it gives the photon back, by emitting light. This is called light emission.
There are several ways an atom can emit light: we can throw UV light, for example, and receive blue light back. This is a special process that happens when an electron jumps, let's say, from level 1 to level 3. Then it comes back from level 3 to level 2 by not emiting light (it is also possible). Finally, it comes back from level 2 to level 1 by light emission. The resultant light, in this case, is less energetic than the light we had thrown at first. So we throw UV light (more energetic) and get blue light (less energetic).
Step 2 - Understanding light emission in fireworks
The whole process of light emission we just described does not really depend on light. It depends on making the electron jump further to an upper level. This can be done by:
1) Throwing light at a sample of the substance
2) Heating the substance
Heating a substance will work just as fine to get the electron moving. Once it gets to an upper level, it can come back by light emission as well, even though light was not involved in exciting the electron.
So in fireworks what happens is that a sample of a substance is heated, making its electrons jump to an upper level. When they come back, they give the energy back in the form of visible light (which our eyes and brain interpret as colours).