When an electron is bounded by the electric charge of a nucleus, it doesn't have enough energy to "run away" to infinity. Meaning you actually need to give energy to the electron to tear it apart from the atom, creating an ion (that is the meaning of ionizing energy).
Looking at it the other way around, if you have a free electron and you want it to bind to an ion to form an atom, you need to take energy from it. Of course, the total energy of the system should be conserved, so some of that energy is released as kinetic energy of the whole atom (meaning some energy goes to the nucleus) but some energy also goes to light. This is quite a general thing to consider: when a charged particle loses energy, because it is always coupled to the electromagnetic field, it can always transfer energy to the field itself (of course only if the conservation laws hold).
This effect is actually what makes florescent light bulbs/ Mercury lamps/ Sodium lamps (etc..) work. You create an electric voltage so high it ionizes the atoms of gas inside the lamps (the electric energy gives energy to the electrons to "run away" from the atom - sometimes not enough to really ionize but enough for the electron to go to a higher energy state of the atom). Then, electrons recombine with the ions, giving away some of that energy as light with specific wavelengths, depending on the energy state they began and finished.
I hope this explanation helped, you are welcome to ask more questions about it.