As the name suggests the binocular twilight factor is the amount of resolution that you get when viewing in the twilight or the dim light.
In general during day time when the light is more than adequate then the main factor for comparing the binoculars is the magnification of the lenses. During night time the aperture size should be the main factor which is controlling the image resolution.
How does that impact you?
For a simple reason that most birding enthusiasts as well as hunters use the twilight zone that is before sunrise and after sunset for their activities.
The twilight factor on mathematical basis is calculated as square root of the diameter of the objective lens and the magnification power of the lens.
Image Credit Gustty
That means a 7×50 binos will have a “twilight factor” of 18.7 and 10×50 will have the “twilight factor” of 22.4
The twilight factor will suggest here that you will be able to see better with a 10×50 in the twilight conditions.
However nowadays with a good coating available the twilight factor is now not a major thing as before. You can have good coatings which will help you get more light in.