My brush color was set to white and changing it yielded no differences.
If you use a black and transparent brush texture, on the object on which you paint you’ll get black from the texture black areas, and you’ll get no color from the texture transparent areas, whatever color you pick on the palette.
If you use a black and white brush texture, on your object you’ll get black from the texture black areas, and from the texture white areas you’ll get the exact same color you picked on the palette.
So in both cases you can’t get any other color than black from the texture black areas. Actually in the video he doesn’t get any other color than black from the texture black areas, he picks red in the palette and gets red from the texture white areas.
The Texture Brush will project what's on your texture on your object, if you choose something else than white on your palette it will just mix the color you chose with the color of the texture.
The Texture Mask brush (below the Texture Brush) will act like a mask, the blacker your texture color is the less the color you picked will be painted on the object.