I know that even to this day disabled individuals are often discriminated against and are treated unfairly. At what point in time, were disabilities first understood at a broad societal level? What cultural changes facilitated this?
The modern culmination of recognition for disabled people in modern times was the Americans With Disabilities Act of 1990, which mandated (among other things) "reasonable accommodation" in public places for disabled people, while prohibiting discrimination against such people. (Variations of the ADA have been transplanted to other countries.)
The ADA was passed by Congressional leaders of the so-called Silent Generation, such as Senator Edward ("Ted) Kennedy. (A few years earlier, a similarly led Congress voted to give compensation to Japanese-Americans interned during World War II.) This was the "Civil Rights" generation (of Martin Luther King, Bobby Kennedy, etc.) that first made their mark as young adults in the 1960s. This was a time of widespread and growing sympathy and understanding for traditionally disadvantaged groups such as women, minorities, and yes, the disabled.