The Ethereum transaction enables transactions with third-party smart contracts. When interacting with these smart contracts for the first time, you must grant permission to those smart contracts to execute trades with your digital wallet. The Allowance Permission is what grants your digital wallet the right to facilitate a transaction with your tokens.
On AirSwap, we set the allowance to a high value, so you don't have to approve your tokens to be traded every time you want to trade.
Below is an example of what this approval will look like in MetaMask: