You are effectively asking about the entire colony-centered history of South Africa, which obviously is too big to relate here. I will say that a new edition of the late Leonard Thompson's very readable A History of South Africa is out, and I would recommend it. His interpretations are a little bit dated on some things, but overall it's an engaging book, and not too expensive either. I will however briefly focus on the "why" here, as you requested.

The initial purpose for both occupations (of the Cape of Good Hope and its hinterland at first, not "South Africa" generally) was useful access to provisioning for Europe-to-Asia journeys. Without a Suez Canal, you had to round Africa for anything significant, although the British did send some communications overland near Egypt after the Napoleonic Wars.

The VOC (Dutch East India Company) set up the station in 1652; the British took it over twice (1795 and 1806) before taking it in payment of outstanding Dutch war debt in 1815. The colony was bigger by that time, with an influx of settlers from western Europe, slaves and transportees from Asia, and the emergence of various mixed communities from among Khoesan and colonial populations; until the 1770s, local populations were rarely large enough to pose a serious impediment to settler acquisition of land, and the Company didn't much care to stop them.

But the capacity of Cape Town to provision vessels was useful to the British too, even if they had to contend with inland settlers who spoke a creolized Dutch (the Boers, or proto-Afrikaners; see Giliomee's The Afrikaners on this). As for active English colonization of the Cape, that came a bit later--the first big pulse being the 1820 Settlers, who serve a key role in genealogy in SA today. But their purpose was both to create a rural population who weren't as fractious as the old littoral population under the weak rule of the VOC, as well as one dense enough to fight Xhosa-speaking polities that were becoming a serious problem to the colony's boundaries. Within two more decades, the colony had incorporated some Xhosa-speakers as allies and auxiliaries, but the stalling of the line of settlement and various other legal impositions led some of the old settler population to move inland or to the coast south of the young Zulu state; that's the origin of the settler influence in the far eastern, central, and northeastern parts of the country.

But initially? It was about a stopping-off point, and maybe a few high-value commodities (mostly wool before the diamond revolution of the late 1860s).

Decolonization is a hard beast to define. Do you use Responsible Government (1872/1893/1906/1907 depending on which Colony), the date of Union (31 May 1910), the Statute of Westminster (1931), the declaration of Republic (1961), or the end of apartheid and the arrival of one person-one vote democracy (1994) that ended overt internal colonialism? Beyond that, what do you exactly mean "how was it?" How was it what, when, in what way, and for whom?