On the specific wording in question
Relative to the Earth would normally mean the Earth is fixed in place. So days relative to the Earth doesn't exactly make sense.
Better, quoting the exact section linked:
The stars are so far away that Earth's movement along its orbit makes nearly no difference to their apparent direction
If it helps, it's basically the same as saying "relative to the universe".
(similar explanation to the Wikipedia article)
In the frame of reference of a solar day, the Sun and Earth are always in a fixed place - say, the Sun directly above the Earth.
On the other hand, in the frame of reference of the stars (universe), the Earth orbits the Sun once per year. Equivalently, the Sun may seem to orbit the Earth once per year.
If this isn't intuitive, pretend there's someone walking around you in a circle, and you keep in eye contact for a whole revolution. They're moving around you, so you're definitely turning a full revolution - and they're always looking in the opposite direction (= at you) - so they must have also turned a full revolution - and since they're always facing you, you must have appeared to made one orbit around them from their perspective.
So no matter how many days are in a year, since the Sun appears to make a full revolution around the Earth, it essentially "catches up" with one spin of the Earth, which is why there is one fewer solar day than sidereal days in a year.