You cannot directly extract your private keys from a Ledger device. They are stored on the secure element and do not leave the secure element. All unsigned transactions or messages to be signed are passed to the secure element, and then returned to your computer after they have been signed. If you could extract the private keys from the Ledger, then there would be a risk to your keys every time you connected your Ledger to a computer, and the very point of the hardware wallet would be more or less negated.

In order to access your private keys, you must use the 24 word seed phrase which you (hopefully!) wrote down on paper and stored securely when you first set up your Ledger device. You would need to import your seed phrase in to another service such as Electrum or iancoleman.io, and then access your private keys that way. Note that this should only ever be done in an emergency, as exposing your seed phrase to other wallets or websites immediately compromises it and puts all your funds at risk. If you do need to recover from your seed phrase, you should do it on an offline computer and immediately move all your funds to a new wallet.