See the Arduino portion of this example for information on how to connect the Arduino Uno and the Beaglebone Black (including the logic level converter).
To configure the UART/serial port on the Beaglebone Black, see this post.
This program reads the data sent from the Arduino via the serial connection. The RFID card data from the Arduino begins with a ^ and ends with \r\n. As the code stands for now, the Beaglebone Black responds to to the received data by sending the card number back with the status message OK and the current date and time. The application will ultimately need to check the RFID tag number against a database and record the use of the tag, but this example is really just intended to show an example of serial communication.
serial = serial.Serial("/dev/ttyO1", baudrate=38400)
resp = ""
while (serial.inWaiting() > 0):
data = serial.read()
resp += data
if "\r\n" in resp:
now = datetime.datetime.now()
timestamp = "%02d/%02d/%d %02d:%02d:%02d" % \
matchObj = re.match(r'\^([0-9A-F]+)\r\n', resp)
resp = ""
serial.write(matchObj.group(1) + " OK: " + timestamp + "\n")