Mint and burn is shorthand for the mechanism described under stablecoin minting and burning: new tokens are created, or minted, when a customer pays in the backing asset, and tokens are destroyed, or burned, when a customer redeems them. The two operations keep the number of tokens in circulation equal to the assets held behind them.
For a fiat-backed stablecoin, a customer wires dollars to the stablecoin issuer, which adds them to the stablecoin reserves and instructs the stablecoin smart contract to mint the same number of tokens to the customer's address. In the other direction, issuer redemption ends with the returned tokens being burned and dollars paid out.
The phrase is also used more widely in crypto. Bridges that move a token between blockchains often work on a burn-and-mint or lock-and-mint basis, destroying or locking tokens on one chain and creating the same amount on another. Some cryptocurrencies burn a share of transaction fees, or send coins to an unusable burn address, to reduce supply. In those cases nothing is being redeemed; the term describes only the change in the number of tokens that exist.