A burn happens when a certain amount of token is rendered as non-spendable anymore, reducing it's total circulating supply.
This happens thanks to a specific function of the smart contract, known as burn function, that sends the token to a specific address of which no one keeps the private key: all tokens sent there will become useless.
The verification of a successful burning is called "proof of burn": thanks to the transparency features of the blockchain, anybody can track and verify these transactions as the burning address is public.