Like a lot of the symbols we now associate with the Klan this is something that comes about in the 1910s as part of a conscious effort to glorify and romanticize the original reconstruction-era Klan.

It’s important to note that when we talk about the entire history of the Klan we’re actually looking at three more or less distinct organizations. First there was the original Klan of the late 1860s and early 1870s, which was a very loosely organized vigilante movement mostly made up of former confederates. They were one of many similar terrorist groups in this era (although probably the largest and most well known) who were trying to overthrow reconstruction and reinstall white supremacy. This group was not really centrally organized in any way and at least at first, they didn’t have a consistent uniform or set of symbols associated with them. They would’ve consistently covered their faces, but otherwise there are accounts of Klansmen in this era dressing outlandishly, sometimes even in women’s clothing. Think more along the lines of drunken frat boys raiding the closet for a night of causing chaos, less noble religious order of knights. This incarnation of the Klan is strictly outlawed and crushed by federal troops in 1872 and there’s essentially zero Klan activity for the next 40 years.

Now by the early 1900s, the political factions that had been anti-reconstruction had regained enough political and cultural capital that they were able to create a popular resurgence of interest in the civil war era, around the narrative that the war had been a tragic and unnecessary misunderstanding, that the confederacy had been a noble “lost cause,” and that reconstruction failed not because it had been intentionally dismantled (and in some cases literally violently overthrown) but because actual multiracial democracy had been corrupt and incompetent. This is the era when books like Gone With the Wind become extremely popular, which portrayed a very romanticized idyllic idea of pre-war southern society, focused on styling the planter class as a European-style aristocracy.

Another hugely popular book from this era was The Clansman by Thomas Dixon Jr from 1905 which portrayed the Klan in a very heroic light, saving the white women of the south from horrible black misrule. This book, and especially its 1915 film adaptation Birth of a Nation, was a genuine pop culture phenomenon and had enormous impact in reviving interest in the klan. The first ever description of burning crosses being used by the Klan comes from Dixon’s novel, and there’s no evidence of the historical first Klan ever actually using this symbol. Instead, Dixon is referring to an old custom used in the Scottish highlands, where small lit crosses carried by men on horseback were sometimes used as beacons to warn people when a battle was going to start. Dixon even has the characters in the novel introduce this as “the old Scottish rite of the burning cross.”

What’s going on here is that Dixon is consciously “updating” the racism of the Klan for a new era. He’s asserting that they have a racial consciousness not just as white people but specifically as part of an Anglo-Saxon Protestant lineage tracing back to Britain (in this case specifically to Scotland where most white southerners would’ve been able to trace their ancestry to). This is in the context of one of the biggest immigration waves in American history, where there’s intense animus now not just at black people but also at poor immigrants from Italy and Eastern Europe, most of whom are Catholic or Jewish.

In September of 1915 a new organization calling itself the Ku Klux Klan is formed, explicitly modeling itself after the romanticized heroic klansmen in the Birth of a Nation movie. Unlike the first, loosely organized Klan, this is a top down organization where members paid dues to the national organization in exchange for the more recognizable white hood and robes uniform, based directly off the costumes in the movie. They also adopted cross burning, although in this time it was just as likely to be a smaller handheld cross as depicted in the movie and novel, rather than the more recognizable large stationary one. This version of the Klan was heavily centralized and basically operated like a multi-level marketing scheme with a lot of emphasis on recruiting new dues-paying members. Unlike the old Klan which was limited to the south, this version was nationwide with a particularly strong support base in the Midwest. While they were enthusiastic about segregation and racist discrimination against black people, most of their rhetoric at this time is directed against immigrants, Catholics, Jews, and (in the west) Chinese. This version died out during the Great Depression when many rank and file members let their dues lapse and the national organization went bankrupt.

The third Klan, and the one that still exists in some form began after world war 2, initially formed out of whatever local chapters of the 1920s klan were still around. In many ways they had more in common with the first klan, being primarily restricted to the south and almost entirely focused on opposing the burgeoning civil rights movement. They continued to use the robes popularized by the second Klan and maintained the local chapter structure, although this time without a single national centralized organization coordinating different branches. They also burned crosses, this time almost always large ones. While the 1920s Klan had also used burning crosses for show in their own private meetings, the new Klan used them almost exclusively as a threat. This version was a lot more openly violent than the 1920s Klan and during the civil rights era went back to the roots of enforcing white supremacy in the south through terrorism.

As for why an ostensibly Christian organization would be okay with a burning a cross… that’s more up for interpretation. I think the key point is that it’s less about destroying the cross and more about the visual of flames in the shape of a cross, perhaps representing a sort of religiously inspired fury. The French Catholic nationalist organization Croix-de-Feu used similarly imagery for the same reason and historically there were and are a number of religious organizations that have a cross juxtaposed with a flame as their logo with no intended racist implications. In the past the original form of a small cross being used as a torch would have also had a practical use as a practical light source as a beacon, especially in medieval Scotland where fire might be the only light source available

Although more can always be said, /u/Georgy_K_Zhukov has a thorough answer to this question here.