For the type of leds you have you don’t need the controller, that’s for controlling each line of RGB leds or white leds with multiple color temperatures (which would then have 3-4 connectors)

Does the controller do anything special (cell phone app to turn on and off, matter, dimming, etc)? If not then you can skip it. If yes then connect the power supply to v+(red)/v-(black) on the 2 pin side and connect the LEDs to the 4 pin with red going to v+ and black going to any of the other pins (see config 1 on box)

Extra info for nerds: these are switched ground (v-) leds. On an RGB strip or a strip with adjustable white you’ll have extra connectors (RGB+ or ww cw + and this setup is called common cathode). The colors are created by mixing and turning off/on color lines. Dimming is created by something called pulse width modulation, which basically flickers the light at a certain frequency. So to create orange on an RGB strip the red diode is held on and green is modulated at 30-50% cycle. The flickering is fast enough that your eye sees it as a solid but dimmer green and because the diodes are encapsulated together you just see orange.

Yours can probably dim via pwm on the controller but they cannot adjust color, so you just need 2 lines. If you don’t care about dimming or potential other features from the controller then you can skip it