It has a tiny electronic chip inside, and a coil made from very thin copper wire. The readers emit a high frequency magnetic field which induces enough power in the coil to turn on the chip. The chip and reader then communicate briefly, and the backend computer system can read the card ID, see that it's linked to you, and which privileges you have.

Edit: it's called NFC, for near-field communication. You can Google it to learn more.

You can probably see the circuitry if you hold the card up to a very strong light source, such as the lens of a projector.