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.

The card contains a tiny antenna and chip. The scanner emits a field that both communicates with the chip in your card as well as provides it power, similar to how a wireless phone charger works.

The chip contains a code that is linked to you in the school's student database. The scanner will check with the database to make sure you are allowed in that location. It may also log what time you scanned your badge and where.

The card has a circuit with an antenna and small programmable chip with a unique identifier but no battery. The reader is actually just a big RF antenna, when you bring the card near enough to the reader the circuit is powered by that RF energy (similar to how wireless charging on a phone works for ease of explanation) enough to produce a small return signal. That return signal contains your unique identifier and is picked up by the reader which then communicates over the network or internet to a server. The server checks if you have access to that door and if you are within your allowable access schedule, and triggers a magnetic solenoid that releases the catch on the door allowing you to open it.

It must have some kind of chip inside it that stores my information so it can identify me when I hold it up to the scanner.

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