A second map shows the racial population distribution as of the 1970 South African Census. In hindsight, the student's confusion was obvious--if non-White South Africans were only allowed to live in certain locations in the eastern part of South Africa, why were there such high concentrations of non-Whites all throughout South Africa?
The simplest response, but one that is least informative, is that there were traditional nomadic groups, primarily in the West, that were allowed to continue their ways of life. They did not own property, and interacted very little with other segments of the population. They largely resided on lands that were otherwise not very useful to White South Africans, such as the Kalahari Desert, as shown in this third (land use) and fourth map (vegetation).
However, arguably a far better answer, in the context of the Apartheid regime instituted in 1948 by the National Party, is that high populations of non-Whites were allowed on White lands because of their use as cheap labor. At first glance, it might seem impossible to have such a high population of non-Whites in such large territories of South Africa, while the National Party maintained their tight grip on the country, and were to be able to impose their brutal, racist ideology. However, this same pattern was seen in the United States under slavery.
In order to demonstrate this I constructed a county-level map of Mississippi from 1860, using a GIS shapefile that was generously sent to me by the state GIS office (at the time of writing, their automated web site was not working--I notified them of the problem, so it may be fixed by the time this analysis is posted). Merging this file with the 1860 Census county-level data, produced the following map of Mississippi. As can be seen, a number of counties were over 75% slaves--the numbers shown in each county represent the total population in that county that were slaves in 1860. In fact, 55% of the total population of Mississippi in 1860 were slaves. With this example in mind, it should be easier to understand how such a large section of South Africa in 1970 could be non-White, when only Whites were allowed to own property in, or even legally be in (without a work permit), 90% of the country.