Yes, the British were the victors in the Anglo-Boer War, but the Treaty of Vereeniging that ended the war in 1902 provided that in the Transvaal Colony and Orange River Colony, "as soon as circumstances permit, Representative Institutions, leading up to self-Government, will be introduced." This implied full "responsible government", as existed at the Cape, where the colonial legislature would be elected and the executive would be responsible to the elected legislature. It should be emphasised that this contemplated a franchise limited to white voters - indeed the Treaty of Vereeniging provided that the British would not enfranchise black people in the Transvaal or ORC before self-government was reached.

After the war the position of High Commissioner for South Africa and Governor of the Transvaal and the ORC was held by Lord Milner (Alfred Milner, 1st Viscount Milner). Milner and his "Kindergarten" of colonial administrators did want a British-dominated South Africa. They aimed for an Anglicization of the former Boer republics, with schemes of British immigration and enforced English-language education; and they intended to withhold full self-government until British outnumbered Boers. However in 1905 the Liberal government of Henry Campbell-Bannerman came to power in London; Campbell-Bannerman and much of the Liberal Party had opposed the war and were considered pro-Boer. They decided to grant self-government rapidly, and elections were held in the Transvaal and the ORC in 1907. In both cases a Boer-led government was elected; the Het Volk party in the Transvaal and the Orangia Unie in the ORC.

It was on such a basis that South Africa proceeded to Union in 1910. Under the South Africa Act (the Union constitution) South Africa had a fully elected Parliament (elected by white voters only, except in the Cape Province) and a government responsible to that Parliament. Het Volk and the Orangia Unie joined with two Cape parties, the South African Party and the Afrikaner Bond, to form a government and a party under the name of the South African Party (SAP). The first Prime Minister was Louis Botha, who had been a Boer general in the war but had reconciled himself to a self-governing South Africa within the British Empire.

Within a few years there was a split in the SAP. The group under Botha and Jan Smuts, who stood for unity between Afrikaner and English, and remaining within the British Empire, remained the SAP. The group under JBM Herzog, who stood for Afrikaner nationalism and ultimately republican independence, formed the National Party. In 1924 the Pact government, a coalition between the National and Labour parties, came into government, pushing the SAP into opposition.

In 1926 the Balfour Declaration (not the one about Palestine - same Balfour, different Declaration) declared South Africa (as well as Canada, Australia, etc.) to be of equal status and no longer subordinate to Britain. This was formally made law by the Statute of Westminster 1931.

The orange-white-blue flag was adopted in 1928 based on the belief that the Prinsenvlag was raised by Jan van Riebeeck when he landed at the Cape in 1652. The inclusion of the mini flags in the middle was a compromise. The adoption of this flag can be seen then as the result of two factors: the election in 1924 of a South African government that was Afrikaner-led and anti-Empire; and the status of South Africa as an independent dominion, albeit still within the British Empire for now.

(Incidentally, you may be interested in some of the other, weirder, flag proposals from the 1920s.)

A good, readable account of the Anglo-Boer Wars and the formation of the Union of South Africa is Martin Meredith's Diamonds, Gold and War.

What u/ctnguy points to is basically it. I would add a few points, however: First, the open hostility of Milner's Kindergarten to Afrikanerdom--they hoped to flood it out with new waves of 'loyalist' settlers, particularly in the Transvaal--did a lot of work to coalesce a unified Afrikanerdom that had not existed as such before. Most Afrikaners, after all, had remained neutral or loyal to Britain during the actual conflict, and had circumstances and pasts that weren't the same. Afrikaners, like the emerging Afrikaans language, were a highly variable group; if one adds the other major cluster of Afrikaans-speakers, Coloured South Africans, you can see that defining this cultural and identic entity was hardly an easy task--and in terms of battles over the language, it's still unsettled.

The process of Dutchifying or Afrikanerizing the flag touches on the deployment of supporters from outside the proverbial laager as well. Beyond the rushed Responsible Government decisions, part of why Het Volk won so decisively in spite of the presence of Johannesburg was the Milner-era decision to bring in Chinese laborers on limited contracts. This pushed a lot of otherwise Anglo-friendly voters into the Het Volk camp for pragmatic reasons. This was less the case in the ORC, of course, but Milner and his fellow travelers really sorely underestimated the power of their assault on the Boers after the war (and by extension the emerging idea of "Afrikanerdom"). The alignment created for Het Volk, incidentally, would repeat in worker anger with the SAP after the Rand Revolt of 1921-1922 to help push Hertzog across the electoral line.

For the flag, it was initially quite a bit different from the one that emerged in the Hertzog government proposals. This is an image here, with the seals of the four provinces. That reflects the key, pre-Balfour (which marks the creation of a SA foreign office portfolio) and pre-Hertzog, markers of loyalty to the Empire. The 1928 flag of course includes the British flag (for the Colonies) and the two Republics on the Prinsenvlag-evoking background. It said "We're not English, but not really Dutch anymore--just in heritage--but definitely not of English or even British affinity as a nation." I'm not sure how the many Irish and Scots SA citizens felt on this, but there are books on the Scots in SA if you're curious and John Lambert wrote a lot about English identity within SA as a counterpoint.

There are a few things that go with this era which are worth pointing out. In 1925, Afrikaans gained equal status with Dutch in terms of the South Africa Act, and was interchangeable, thus giving Afrikanerdom its own recognized tongue at that level. Likewise, the growing political unity of the Afrikaner bloc in ways u/ctnguy indicated was a vital element, because this flag was a way of marking them as the 'real [South] Africans' as opposed to the "English" who were somehow lesser and, possibly, ephemeral based on their reduction within the 1928 flag.

For the Afrikaner part of this, I'd point to Hermann Giliomee's The Afrikaners: Biography of a People 2nd ed (2009). Meredith, which is mentioned, is a readable introductory work to the background, although there are of course other works on the 1902-1924 era if one wants to get into the weeds. For a general history, there's a new 4th ed of the late Leonard Thompson's A History of South Africa, done by one of his former students, Lynn Berat. But if you want to get lots of bibliographic leads to follow up in English and in Afrikaans (many period) nothing has yet superseded Davenport/Saunders 5th ed. South Africa: A Modern History (2000) but I hope someone takes up the task one of these days. That's a late-career project, though, so I'm not sure who might.

(And u/ctnguy, great to see you--hope you are holding up well in the beautiful lands.)