Is it permissible to take items left behind at the airport, and what is the ruling on someone who has previously taken something from them, and how can they atone for that?

You must adhere to the airport regulations, which stipulate handing over lost items to airport security, as Allah Almighty says: "O you who have believed, fulfill [all] contracts," and because doing so achieves the Shariah objective of returning lost property to its owners. Your taking any part of it is a forfeiture of others' rights and a sin.

If what you took was trivial, something its owner would not pursue, then there is no objection to keeping it if you fear harm from handing it over to security. However, if it is valuable, you must hand it over to security because it is the only place where the lost item can be identified. You are religiously obligated to identify it yourself or to hand it over to a trustworthy person who will identify it, as stated in "Bulghat al-Salik": "It is obligatory for the one who found it to identify it himself or through someone he trusts due to their trustworthiness, and there is no guarantee upon him if he hands it over to a trustworthy person who identifies it."

If you are quite certain that the owner of what you took in the past will not inquire about it, then you must donate it or its value with the intention that the reward goes to the owner, and do not repeat such an act.