How can I color my Mandelbrot set like this?

Asking the original author for their exact Ultra Fractal 3 parameters would be my first choice.

Mandelbrot fractal images are often made by selecting the color based on the iteration where the iteration "blew up" or the point escaped (modulo some factor, perhaps, that keeps the number in the color map (and I don't know how deep people let these iterate nowadays)) for some arbitrary definition of "escape". For $$ z_{n+1}=z_n^2+c, $$ and $c=(x,y)$ of the point you're testing for inclusion. At the very least you need the coordinates of the four corners of the domain (or anything equivalent) and any parameters that might map $n_{rm blowup}$ back into the color map and the color map (which you can probably pick off an image with your favorite image manipulation program (like Photoshop) and the eyedropper tool). You need your own (their) definition of "blow up" since you can't iterate forever. It might just count as leaving the domain or 10x the domain or $n$ too big or $ldots$, but it's been since high- school since I coded one of these (in QuickBasic(!) as a language learning project).

I think your best bet is to reach out to the website owner and see if they'll help you out.