In Indonesia, reducing fuel subsidy strongly impacts the price of everyday goods. Why?
Consumers rarely have to pay the full price of petrol / gasoline. That's true in most of the world, and the US is no exception. There are explicit subsidies and favourable tax regimes in many extraction countries; and of course there are the implicit subsidies in the unpriced pollution - oil's externalities.

Those explicit and implicit subsidies have been there for decades: and consequently, oil products are hugely over-used, compared to the alternatives: that's a direct consequence of oil being a product with demand that's not perfectly inelastic - indeed, estimates of long-term price elasticity of demand for petrol are of a figure around -0.6 or so. That means that oil is embedded in large quantities in the supply chains for very many products.

The larger the historic subsidies, the bigger the impact of changes in subsidy levels to the price of other goods and services. And the poorer the population, the bigger the impact. That's why you see a bigger impact in Indonesia than in the US.

Now, although in the long run removing some of the subsidies will correct some of those imbalances, that will require capital reallocation and some industrial restructuring, all of which takes time.

And although in the long run, rising prices of some products, in the absence of rising wages, will choke off demand for other products, leaving inflation at no net change, the short-term effect of rising oil prices is to raise the costs in a significant part of the supply chain for a lot of goods and services, given an instant boost to inflation.