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.