You need a 12v power supply and some MOSFETs which will act as the interface from the low voltage arduino board to your 12v LEDs. You have to be careful about which ones you buy, there’s all kinds of fun stuff to look at like gate threshold values and voltage differentials, etc. I initially bought some that said they would work with Arduino but it ended up not working for my esp32 because it only outputs 3 volts as opposed to 5 volts and it wasn’t enough to switch the mosfets. These worked for me and should work for you:
Https://www.amazon.com/N-Channel-Power-Mosfet-RFP30N06LE-220/dp/B071Z98SRG
You will need one per color per strand. So if you using the rgbw strip you’ll need 4 per strip. And here is how to wire it: