Everywhere a question like this is asked, I see people responding that (in a scenario where a card is used) the card does some processing with the data it receives/generates some data when it receives a signal. How is this possible without power?

And even if that's the case, why can't every NFC tag in, let's say, credit cards just be cloned because there are no variables in them and only static data? You'd think those RFID tags could be copied and used for transactions.

Because the cards contain a chip which are powered by a coil.
The coil is not really a antenna, but half of a transformer.

Think your regular mobile charger. This contains a transformer, that will transform the voltage from 230V or 120V AC to 5V DC. This is done by having a coil magnetize some iron, and this iron magnetizes the "receiving coil". If you draw current from the receiving coil, the primary coil will also draw more current.

Now, let's go to the "passive" card again.

The reader is one half of a transformer, and the card is one half of a transformer, but this transformer does create a magnetic field in the air instead of magnetizing iron.
When you put the card close to the reader, the reader and card becomes a full transformer, and thus the card can be powered, like it was connected to a battery.

For the reader to transmit information to the card, the reader only needs to vary the frequency or amplitude of the AC voltage that powers the primary coil. The card can sense this and act on this information.
For the card to send information back to the reader, the card simply short-circuits its own antenna via a transistor and a resistor. This will, like the mobile charger, cause the primary coil, i.e. the coil in the reader, to consume more current, and the reader can sense this (by having the primary reader coil via a resistor and then measure the voltage over the resistor) and read the data the chip sends to the reader.

This means that half-duplex bidirectional communication is possible with RFID, thus the chip can do anything, and work like a contact smart-card.
And as you know, a contact smart card with a security chip, that can securely store a key, and only perform operations with the key, is impossible to "clone" or "copy" as the key cannot be extracted. That's the security of smart cards, they cannot be cloned, and that's why they are preferred over magnetic strip cards.

Thus, the same applies to wireless/contactless RFID card.
