Why does dark energy seem to push things apart? Ordinary matter acts to slow down the expansion of the universe. That makes intuitive sense, because the matter is exerting a gravitational force, acting to pull things together.
Moreover, dark energy has “negative pressure” and negative pressure corresponds to a pull. Then why are galaxies being pushed apart?
 A: When treated properly, one has to study Einstein's equations with the dark energy as the source. For example, when the dark energy is cosmological constant (the most important subtype of dark energy), the resulting spacetime is a de Sitter space, a Minkowski-signature type of a hyperboloid. Galaxies move along geodesics and those geodesics are "repelling" each other, like those on an ordinary hyperboloid.
Alternatively, I believe that the following heuristic picture in "classical material science" is an OK qualitative explanation. You may indeed imagine that the Universe with the negative pressure is like a stretched rubber band that wants to shrink. However, the right way to think about the galaxies is that they are not glued to this rubber band. Instead, they are free.
So the result is that due to the persistently stretching rubber band – the space – the underlying space is exponentially shrinking. But the galaxies sit at "fixed positions" – which they may because the underlying space is still effectively flat – so the proper distances between them exponentially grow because as the rubber band shrinks, there is an increasing amount of rubber (proper distance) per unit coordinate length (per the separation of the two galaxies).
