As the country’s rich elite took off in their private jets, the “revolution”—sparked in part by hiked up fuel prices—showed no signs of slowing down.

The real story being buried here is how oil-rich Kazakhstan has high oil prices. This article explains it a bit better.

TL;DR Years of government interference in setting fuel prices.

Domestic producers weren't allowed to set prices above their costs, so they could rarely make a profit, causing a gradual breakdown in their domestic oil industry. But when the government finally switched to a market-based pricing system, they did it too quickly and it caused a massive shock that resulted in huge price increases due to high global demand along with domestic producers nearly on the verge of bankruptcy and low production due to not being able to maintain their facilities for years.