Their entire mindset is based on the presumption of SEC football dominance, which is drummed into them constantly by their program, their opponents, and the TV network they watch. Of course, SEC superiority in the CFP era has nothing whatsoever to do with South Carolina. It's always been them boasting about the achievements of the teams that crush them on an annual basis.
They nourished themselves through all those 5-7 seasons on the ESPN-backed assurance that in any other conference, they'd be 8-4 or 9-3, just a rung below the elite programs (in our case, OSU, PSU, Michigan). Finally(!), a year comes where though they're not invited to the party, they're in the conversation. They get put into a bowl, not with a B1G blue-blood like Michigan, but a program for which they have no regard, or even awareness. They never for a moment anticipated that they could lose this game. Opt-outs? Who cares? Throw 11 guys on the field with SEC patches on their jersey and watch a traditional B1G bottom-dweller melt away. This was their (one?) shot to prove that they were always much more than what their records said they were.
When they realized that we were outplaying them, it wasn't a disappointing end to what was a fine season for their program. It was the harsh realization that all those years when they finished ninth in the SEC, the fact that Alabama or Georgia beat the hell out of Ohio State or whoever didn't make them any better than whoever finished ninth in the B1G. That's what was on the faces of their players. That's what brought Beamer to incoherent rage on the sideline. That's why most of their fans online refuse in any way to deal with what actually occurred between the lines but are obsessed with what, at worst, would be a mild tease using an innocuous officiating signal (though I know it wasn't that).
Everyone loses games, but SC lost the narrative they've been clinging to for decades. That's why their brains have short-circuited.
Being much closer to the situation than probably everyone else here (but also not a dyed-in-the-wool, born-and-bred Gamecock fan), I am able to confirm/admit that much of this is 100% true. It would absolutely be fair to say that South Carolina has done a lot more riding of the proverbial waves than creating them, and fans are often guilty of puffing their chests a bit too much based on a conference prestige they've had an undersized hand in cultivating.
South Carolina is obviously not among the SEC elite, and hasn't been at all responsible for the SEC winning over half of the national championships this century (as well as over half of those in the CFP era)...but the facts embedded within this statement do help to explain the mindset a bit.
South Carolina has not exactly been a doormat, over that time period. They've finished with at least 8 wins ten times since 2000, including a string of three seasons in row with 11 wins, from 2011-2013. And, yes...over that same time period (prior to Tuesday's game), they were 6-1 against the B1G, with wins over Ohio State (2x), Michigan (2x), Wisconsin, and Nebraska (the only loss coming in a drubbing against...Iowa

).
And there are other factors which help to fuel that "it just means more" attitude (and cockiness). Sellouts at South Carolina are more the norm than a rare exception, and Williams-Brice Stadium is often cited as one of the toughest road environments in the nation. Objectively, I would put the atmosphere of a sold out night game there up against just about any other in all of college football. The fans are passionate and loyal to a fault, and I would argue that they're hardened by a lot of the same kind of seeking for respect that Illini fans know all too well (just on a somewhat different level). Clemson's recent successes have certainly not helped in this regard!
All of this to say: if you put it all into perspective, Illinois didn't exactly present much of a threat or much of a tantalizing opponent, especially with the way the Gamecocks finished out the regular season and all of the talk about the Playoff (right or wrong). The recent history of Illini football is what it is. But, as SuperintendentChalmers aptly pointed out, it made for quite a shock to see the Illini outplaying the Gamecocks, and ultimately winning the game.
It's a necessary (and, in my opinion, very sweet) component of rising from the ashes, as a program. Perceptions will be slow to truly change, nationally, but games like this go a long way. Enjoy the ride! I-L-L!