Download the MFRC522 repository as a zip file and extract it into your arduino libraries folder. For some reason the official releases don't contain the MFRC522hack and MFRC522debug classes, which we need.
Connecting the RFID module to Arduino is described in "MagicMifare.ino"
Open the "MagicMifare.ino" in Arduino IDE and upload it to your board. Then open the serial monitor.