What exactly is going on here?

Cryptocurrency was made famous by Bitcoin, but Bitcoin is designed just to transact and record ownership of Bitcoin itself. NFTs are minted on a successor to Bitcoin called Ethereum which allows for arbitrary computer code to be executed on the blockchain and the results of the execution to be stored forever.

Originally, the digital images were too large to store on the blockchain, due to some clever compression ideas and some help, each PNG image in a collection and its corresponding attribute data are recorded fully on chain. You can read more about the mechanisms and details of the process on Wikipedia.