EOTF = electro-optical transfer function
It's the technical term for what people are talking about when they say things like "tone mapping" and "custom gamma". It's the function that transfers/maps the full luminance range of the HDR signal to the capabilities of the display.
So for example the HDR content may be mastered at 1000 nits peak brightness, but the projector can only do 150 nits peak brightness off the screen. Somehow you have to translate the full dynamic range of the HDR signal to the lesser range of the projector.
One option is to just squish it all in with a direct linear transfer, but that will result in a very dark image since squeezing in all the dynamic range of the highlights will necessarily compress all the mid/shadow detail too (this is why many people complain that HDR looks too dark, and probably similar to what Gamma D does).
Alternately a custom EOTF curve, whether programmed by a calibrator or done automatically in the player or display, will "shape" it more intelligently to preserve the dynamic range of the shadows and middle regions and then roll things off more steeply in the highlights. So the image doesn't look dark and gross but you still maintain some detail in the specular highlights regions. The "HDR brightness" slider control you'll see on many players and displays basically changes the shape of this transfer function, but it will be a compromise in one direction or the other. If you try to preserve more highlight detail, then by definition you'll have to squish the dark/mid areas of the image down so the overall image will look darker. If you want the overall image to be brighter, then the inherent trade-off will be losing some of the detail in the highlights.