You can translate as " about", maybe the meaning is: (He) had reason (about that someone says that) love does not exist.

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Regarding when to use the expression, it is commonly used when referring to a saying or even something someone has said. For example "el amor no existe", "el trabajo dignifica al hombre", "el dinero no hace la felicidad" are very common Spanish phrases.

Are you sure your sentence should say, "tenía razón eso de que ....." and not "tenía razón sobre eso de que...."? If you add the word "sobre", it makes perfect sense: "He/she/you was right (correct) about the fact that love doesn't exist." Silvia's sentences would be translated, "The fact that you wrote the letter is ok", and "The idea/fact that I can't do it is a lie." In general "eso de que" means "the fact/business/idea/thing that". Example: No creo eso de que yo no puedo aprender ruso." I don't believe the idea/notion that I can't learn Russian. Or, "I don't believe that business about my not being able to learn Russian."

Examples: Eso de que tú escribiste la carta está bien. Eso de que yo no puedo hacerlo es mentira.