On the software side there is (almost*) no magic in the Tangem cards. It is just a normal wallet without providing a backup of the seed words.
It is fine to know how they sign transaction, they do it like every other wallet - because, like every other wallet, they need to implement Cardano specs otherwise the network nodes would reject the transactions.
*: almost because theoretically Tangem cards support reading out the amount of the card without being online. Therefore the store some kind of magic value directly on the card when creating / loading the wallet the first time.
This feature would make offline transactions possible (by handing over the card) but is currently not available for Cardano cards. And this is the main reason why the card don’t give you a backup seed. You do not want the person who paid you with the card to later restore the wallet via the backup seed words.
Check out my comment here: Cardano Tangem Cards: What are they and how to use them? - #3 by adatainment