Can I control/read an RFID reader with an Android device or do I have to put an Arduino board between the two?
Android devices can usually connect directly to USB devices in OTG mode. It looks like that board has a USB serial interface, so it should work fine if you connect it with the proper USB OTG cable.

Now, how you actually talk to it is a different story. I'm far from an expert on Android, but a quick Google search turned up https://github.com/mik3y/usb-serial-for-android which may do what you need.