The photographer is referring to 'expose to the right'.
This means you expose as bright as possible without overexposing / clipping highlights.
(Note: in some cases one can choose to overexpose certain things, like bright skies or reflections or even a white background that one wants to be pure white.)
This is called 'expose to the right', because this method puts as much as data as possible on the right half of the histogram. (On the left corner: black and on the right corner: white.)
I've been told it his has something to do with the logaritmic properties of storing light as data. (Techno-blabla-stuff

)
Anyway, this is done because brightening the image (or parts of it) introduces grain or noise to the image, while darkening does not.
I like to shoot pictures as bright as possible as well. Not just for technical reasons, I like the look as well
However: for a photographer it's much easier to 'expose to the right'. When shooting RAW-pictures some highlight overexposure can even be saved in the raw-editor. Video isn't as forgiving when you shoot a MPEG, MJPG or H.264 codec.
When shooting video I like to expose as bright as possible, but I also try to keep skintones around 70%. This is where zebras are a great tool (that many DSLRs seem to lack).
Under-exposing is something I like to avoid: it only introduce extra noise when correcting it in post.
However: when shooting video with a DSLR you can meter the light.
You know you shuttertime and ISO, so measuring properly will tell you what diafragma (stop) you need for correct exposure. (But is something not a lot of people do for video.)