Any explanation can get arbitrarily more and more detailed and involved, especially when the question is this general. A better solution is probably to find some basic reading material on the subject and ask specific questions about stuff you don't understand.
The short answer is that fire is formed when some exothermic chemical reaction releases energy into the surrounding air. The air heats up, ionizes and becomes a plasma. Light is emitted by several different mechanisms. The gases involved will give off light via spectral emission and black body emission. (For instance, this is why hot soot and embers glow in a fire.) The color of the light emitted will depend on electronic transitions in the constituent atoms (for spectral emission) and on the temperature of the fire (for black body emission.)
As far as formulas go, quantum mechanics and thermodynamics play a huge role in describing the processes involved. You might look into reading up on the thermodynamics of black body emission, and on the basics of photon emission from atomic transitions. (The simple description given in most introductory chemistry books should suffice for a general discussion like this.)