I think you would enjoy reading Curt McMullen's paper "Moduli spaces in genus zero and inversion of power series". In some sense there is nothing there that isn't already in Getzler's paper, but everything is stated in a down-to-earth and combinatorial fashion.
Let me summarize the story, first for the spaces $\overline M_{0,n}$ and $M_{0,n}$. The space $\overline M_{0,n}$ has a stratification where a stratum corresponds to a tree with no vertices of valence two. The stratum itself is isomorphic to $\prod_v M_{0,\mathrm{val}(v)}$ where $v$ runs over interior vertices of the tree and $\mathrm{val}(v)$ denotes the number of incident edges. Since the virtual Poincaré polynomial is additive over stratifications, this shows that the virtual Poincaré polynomial of $\overline M_{0,n}$ is given by a sum over trees involving the virtual Poincaré polynomials of $M_{0,n'}$ for $n' \leq n$. Now using the relationship between compositional inversion and summing over trees, well-known to combinatorists, one can thus show that the exponential generating series of virtual Poincaré polynomials of $\overline M_{0,n}$ and $M_{0,n}$ are compositional inverses of each other. (If you don't know virtual Poincaré polynomials, think about any other invariant additive under stratification, e.g. Euler characteristic.)
Finally, both the spaces $\overline M_{0,n}$ and $M_{0,n}$ have pure cohomology in every degree: $H^k (\overline M_{0,n})$ is pure of weight $k$, and $H^k(M_{0,n})$ us pure of weight $2k$. Thus in both cases, the virtual Poincaré polynomial concides with the usual Poincaré polynomial (in the latter case up to a substitution $t \mapsto t^2$). This explains the second sentence in Bergström-Brown's abstract.
The story for $M_{0,n}^\delta$ and $M_{0,n}$ is completely similar, the only difference being that $M_{0,n}^\delta$ has a stratification indexed by trees without vertices of valence two and with a cyclic ordering of the edges incident to each vertex. In the same way as compositional inversion of exponential generating functions corresponds to sums over trees, compositional inversion of ordinary generating functions corresponds to sums over trees with such cyclic structure. McMullen touches upon something very similar at the very end of his paper. He doesn't consider $M_{0,n}$ and $M_{0,n}^\delta$, but instead considers the choice of a connected component of $M_{0,n}(\mathbf R)$ and its closure. Combinatorially this amounts to exactly the same thing: $M_{0,n}^\delta$ is defined by choosing a connected component of $M_{0,n}(\mathbf R)$ and taking the union of all strata meeting the closure of this component.
A final remark is that the duality between $H^\bullet(\overline M_{0,n})$ and $H^\bullet(M_{0,n})$ can be upgraded to a Koszul duality of two cyclic operads, the "Hypercommutative" and "Gravity" operads. This is a much stronger result than just that their generating series are compositional inverses, and this is what Getzler proves. On the other hand the cohomologies of $M_{0,n}^\delta$ and $M_{0,n}$ give rise to nonsymmetric cyclic operads (this notion is not defined in the literature, but it's not hard to give the definition). However, it turns out that they are not in any natural sense Koszul dual of each other, but it is still true that they are interchanged with each other under bar-cobar-duality, up to homotopy. (But first one needs to define a bar transform of nonsymmetric cyclic operads...) This is an operad-theoretic statement that improves on what Bergström-Brown proved. I worked this out with Johan Alm at one point but we never wrote it down properly.
