This is a quick definition in support of a lesson. I will have more to say later.
A super simple explanation of Tempo Rubato can be constructed out of one word: Jazz.
Most people can understand the willful manipulation of melodic content that Jazz musicians and singers are required to do in order to fit within Jazz styles. The classical singing student is often told to hold strictly true to the notes written on the page. I have heard that one teaching professional I know says that Tempo Rubato doesn’t exist. At least in his studio it would seem to be unwelcome. It is in fact just as necessary to interpretation as it is distrusted by pedants.
Mozart is not Jazz, but Tempo Rubato suits his music well. It is only a matter of degree and artistic choice. Anyone wanting to become a great interpreter will learn to use Tempo Rubato.
The rules are simple. Stay with your accompaniment in the large scale, and deviate in the small scale according to the language and your personality.