We'll deal first with the RGB lighting questions, then get back to fan motor operation.
As jacob68 has pointed out, that connector in your photo is a plain RGB (4-pin, 12 V system) female connector, and it is on the end of a cable that starts from the point labelled "M/B-IN" on the Thermaltake case's RGB control board. You can use the case's own top front pushbutton to cycle though the RGB effects in your front fans. But the case manual also says that IF you connect the second cable (the one in your picture) to a plain RGB (4-pin 12 V system) header on your mobo (IF you have one), than you can switch to using mobo control of those lights. To do this you hold down the case's top button for at least 3 seconds, and then the front lights will blink Red twice and switch to using mobo controls. Then you must have the mobo's RGB lighting control software installed and running - in this case, the Gigabyte RGB Fusion utility.
Your mobo DOES have this header, but it's a little different from what you may have looked for. See your mobo manual p. 13, item 8 and p. 16, "LED_C" header. This header has FIVE pins because it is a variation of the 4-pin plain RGB header that also can be used with an RGBW system. But note that it has one pin labelled for 12 VDC, and the other four pins labelled for different colours. Now look closely at the connector in your photo, and you will find one hole is also labelled for 12 VDC, maybe with just an arrow. Match the 12 VDC hole and pin, leaving the fifth pin of the mobo header unused. Now you have the option to use either the case's pushbutton switching or the mobo's RGB Fusion utility to control the front fan lights.
Now, the case website and instruction manual makes no mention of how the front case fan motors are powered and controlled. I can find only two clues, so I'm speculating here, and maybe the best plan is to call Thermaltake Tech Support and ask them what to do here. But you also can try this.
The "clues" I see are this: the connections to the first three fans from that case fan Hub unit show SIX wires going out to each fan (and three out from the "FAN4" port). Of these, four are clearly for RGB lighting, and the other two are marked "+F-". That looks like those are the + and - power supply lines to each motor. If so, then the connections already are made, and the question is how does one exercise control? Well, that large rectangular button on the case top front has two ends, labelled "P" and "R". Some of their instructions indicate you use the "R" end to make changes to the RGB lights. MAYBE the "P" end is the fan speed selection switch. Try pushing that and see if the fans work.