How do exited atoms give off light

Light is produced by the change of state of energy of an electron. Within an atom, there are multiple electron "shells". 'Excited' atoms are those where the electrons are moving in between shells. The electrons move because they are given energy enough to move to a different shell; however, they are unstable and cannot remain in their new position. When moving back to its original shell, the electron loses energy - but energy can't be created nor destroyed so they release a photon.

There's other ways light may be produced, but always it is produced as a result of transmission of energy (in some form). Also, light is usually produced in conjunction with other forms of energy such as sound or heat.