Is there anyway to test that the external power supply module works with just an arduino, breadboard jumper cables and LEDs. I don't have a dc power supply for the module which is why I'm asking.
Any help would be appreciated!
Ah yes, one of those modules!
What is it you mean "works with"? The Arduino includes essentially the same components as this, so you do not need it for an Arduino. In fact, exactly what this device is useful for is not entirely obvious. It might be useful with some components on a breadboard when you do not have an Arduino.
Both this and the regulator on the Arduino have an important limitation - if supplied by a 7, 9 or 12 V supply via the "barrel jack" - bottom right corner on your picture - it can only provide something like 100 or maybe 150 mA in total without overheating as it has almost no heatsinking. You are far better off to use a regulated 5 V supply such as a USB "phone charger" plugged into the USB port on a UNO or Nano or the "reverse USB" connector on this module which then puts the 5 V - and some 3.3 V if needed - on the common rails of the breadboard.
Can't see the bottom... looks like a breadboard power supply.
Comes in really handy when breadboarding projects, indeed if no complete Arduino board is used, such as a barebones ESP8266 or ATtiny or something without MCU at all.