Well, the explanation really is held in Gauss's Law. That shows that the field of an infinite line is distance dependent, while an infinite plane is not.
But I expect you're looking for a more intuitive answer. So I'll give my best shot at one. Keep in mind, that dealing with an infinite anything tends to be non-intuitive, so this may be slightly "hand-wavy".
Electric field works on an inverse-square law, meaning that the when you view a charge from twice as far away, the field strength is four times weaker. It turns out that visual perception is also (approximately) an inverse square law. The amount of area in your field of view an object takes up follows the same proportionality as electric field does. So a way to get a rough estimate for how an electric field changes strength, is to see how the sources size changes with distance.
If you consider looking at an infinite plane, it would take up your entire field of view, (assuming your view is only 180 degrees). As you got further and further from the plane, it would still take up your entire field of view. So the strength of the field doesn't change.
However, the infinite wire does change size. While it's long dimension stays the same in your view, it get's thinner. So the field strength gets smaller and approaches zero at infinity, since the wire would appear infinitesimally thin as you approach infinity.
Hopefully that made it a little more intuitive!