Any peg (such as a stablecoin) relies on two mechanism to actually stay stable:
Trust. All value is based on belief. If people believe that the coin's value is equal to $1, it is. And if people believe that most people believe that it's worth $1, they are also likely to believe it. It's a self-sustaining cycle of trust and belief.
Large buy and sell orders above and below the $1.00 price maintained by a large organization and/or by people trying to make a profit if the price fluctuates. For example, there might be a huge stack of sell orders at $1.01 and a huge stack of buy orders at $0.99. The fact that people can see these buy and sell orders on exchanges further enhances the trust in the peg.
If you know that people believe that it 'should be' worth $1.00, you are economically incentivized to participate in maintaining the peg by placing a buy order at $0.99 and a sell order at $1.01. You can make a small profit if the price goes up and down a bit. This is 'decentralized and autonomous' in the sense of many different people placing these orders because of their own trust in the peg and the resulting incentive.
Most decentralized and autonomous systems are based on human incentives. For example, Ethereum miners could choose to not mine any transactions any more. However, they are economically incentivized not to do so because they would lose the transaction fees and they would reduce the value of ETH itself.
Related anecdote: When the UK entered the European exchange rate mechanism, the British government tried to peg their Pound to the German Mark. Billionaire George Soros managed to break the peg by selling huge amounts of Pounds. This selling started to exhaust the stack of buy orders, which scared other traders and caused even more selling. The British central bank simply had not allocated enough funds to buy all the pounds that were being dumped and the peg broke.