The surprising discovery that the universe is not just expanding but that the rate of expansion is increasing is what led to the idea of dark energy as the cause of that acceleration. But explaining in non-mathematical terms to a lay audience why dark energy leads to an accelerating expansion is not easy and introducing the idea that dark energy creates a negative pressure does not help much either.
Sean Carroll, one of the best expositors of cosmology to non-scientists, tries to explain what is going on.
For cosmological dynamics, the relevant fact about dark energy isn’t its pressure, it’s that it’s persistent. It doesn’t dilute away as the universe expands. And this is even a fact that can be explained, by saying that dark energy isn’t a collection of particles growing less dense as space expands, but instead is (according to our simplest and best models) a feature of space itself. The amount of dark energy is constant throughout both space and time: about one hundred-millionth of an erg per cubic centimeter. It doesn’t dilute away, even as space expands.
So: the density of dark energy is constant, which means the curvature of spacetime is constant, which means that the universe expands at a fixed rate.
The tricky part is explaining why “expanding at a fixed rate” means “accelerating.”
The point is that the expansion rate of the universe is not a speed. It’s a timescale — the time it takes the universe to double in size (or expand by one percent, or whatever, depending on your conventions). It couldn’t possibly be a speed, because the apparent velocity of distant galaxies is not a constant number, it’s proportional to their distance. When we say “the expansion rate of the universe is a constant,” we mean it takes a fixed amount of time for the universe to double in size. So if we look at any one particular galaxy, in roughly ten billion years it will be twice as far away; in twenty billion years (twice that time) it will be four times as far away; in thirty billion years it will be eight times that far away, and so on. It’s accelerating away from us, exponentially. “Constant expansion rate” implies “accelerated motion away from us” for individual objects.
It is still not easy to understand but definitely one of the better attempts that I have seen.