This is the official quote from a SA government website as to why June the 16th is a public holiday:
Previously known as Soweto Day.
In 1975 protests started in African schools after a directive from the previous Bantu Education Department that Afrikaans had to be used on an equal basis with English as a language of instruction in secondary schools. The issue however, was not so much the Afrikaans as the whole system of Bantu education which was characterised by separate schools and universities, poor facilities, overcrowded classrooms and inadequately trained teachers. On 16 June 1976 more than 20 000 pupils from Soweto began a protest march. In the wake of clashes with the police, and the violence that ensued during the next few weeks, approximately 700 hundred people, many of them youths, were killed and property destroyed.
Youth Day commemorates these events.
This doesn't quite sum up one of the most important days in South African History. It took another 20 years after this day in 1976 for the process of change to really become apparent.
Click here for more information on the 1976 uprising.