There is nothing about this strip of leds that is special to the BBC microbit. They can be used with Arduino also. These leds are ws2812b type, sometimes called "Neopixels". There are two Arduino libraries you can use, called "Neopixel library" and "FastLED library". Each library comes with example sketches that you can use to test the strip and then use as the basis for your own sketch.
However, what the Kitronic page does not show is that you should place a resistor (e.g. 330R) between the Arduino pin and the data-in pin on the strip, and you should also place a capacitor across the power lines to the led strip. For a small strip, a 100uF cap will be OK, but if using larger strips, or multiple strips (for example 4 of the arcs to make a circle) then a 1000uF would be better.
When using larger strips or multiple strips, a separate power supply will be needed. The Arduino (or microbit) cannot supply power for more than around 5 to 10 leds. For example, if you connected 4 arcs of 12 leds in a circle, you would have 48 leds which can draw up to 60mA each, so you would need a 5V power supply with at least 3A capacity. A power supply like this could also power the Arduino.