"Expose to the right" means record the brightest image you can and then reduce the brightness in post to achieve the desired level.

The word "right" comes from the histogram, where conventionally brightness increases left to right, thus increasing brightness shifts the whole histogram to the right.

ETTR helps reduce noise simply by capturing more light, which reduces photon noise, and gives a better signal to [electrical] noise ratio (by virtue of a bigger signal). The reason high ISO photos look noisy is due to low levels of light and amplifying a weak signal.

The technique works provided you don't increase the exposure to the point where it hits the maximum possible value and gets cut off, as this will result in a loss of information (known as clipping/blowing the highlights). Typically this is seen as an area of the image (usually sky) which has gone pure white.

In principle the technique works for film, certainly exposing the left and then having to push your image when printing will increase grain. However film has a different cutoff characteristic, as highlights gently roll off rather than hitting a hard limit.

Here's an experiment I did to demonstrate the effect (and rebuff a blog article which claimed ETTR didn't work):

Here's the camera metered exposure:

Here I've used ETTR and increased the camera meter's exposure by 1 stop using a longer exposure:

Finally, to show the difference here's the standard exposure with the ETTR image offset in the centre:

The reduction in noise is visible, particularly in the purple patch in the bottom left.