In my opinion there are two steps involved in proving the relation. All the proofs I know go as follows: (1) prove that the ribbon graph homology complex computes the cohomology of the geometric realisation of the category $Ribbon$ of ribbon graphs, and (2) show that the geometric realisation of this category is homotopy equivalent to the classifying space of the mapping class group.

For the first step, I like Igusa's proof in chapter one of *Graph cohomology and Kontsevich cycles* (arXiv:math/0303157v1) The idea is that there is a canonical chain map $C_\ast(|Ribbon|;\mathbb{Q}) \to \mathcal{G}_* \otimes \mathbb{Q}$ up to homotopy, where the first is the cellular homology and the second is the compactly supported graph cohomology complex. This comes from an acyclic carrier over $Ribbon$ known as the forest carrier. The idea is sent a ribbon graph to the chain complex of generated by all ribbon graphs mapping to it (by collapsing subtrees), which is acyclic and augmented over $\mathcal{G}_*$.

For the second step, there are many proofs. I know of at least three distinct ones, but unfortunately only on the details of two of these.

(1) Strebel's proof uses the analytic theory of quadratic differentials. A horizontal trajectory of a non-zero quadratic differential is a curve along which the differential attains real positive values. These trajectories are either closed or non-closed. The closed trajectory come in families and these families decompose the complement of the non-closed trajectories into annuli or punctured disks. In a single annulus or punctured disks all closed trajectories have the same length. A Jenkins-Strebel is then one for which the non-closed trajectories have measure zero. Together with the zeroes these from a ribbon graph. However, one can do even better: it can shown that given a surface of a genus $g$ with $n$ marked points such that $2g+n>0$ and $n$ non-zero real numbers, the surface minus the marked points admits a unique Jenkins-Strebel differential such that the closed curves are of the given lengths. Conversely, given a metric ribbon graph, one can construct an essentially unique Riemann surface. This gives a bijection between $\mathbb{R}_+^n \times \mathcal{M}_{g,n}$ and the space of metric ribbon graphs à la Kontsevich in *Intersection theory of the moduli space of curves*, which becomes a homeomorphism when picking the correct topologies. It is then not hard to show that the space of metric ribbon graphs is homotopy equivalent to $|Ribbon|$. Kontsevich gives a nice summary of this.

(2) Costello, in *A dual point of view on the ribbon graph decomposition of the moduli space of curves* (arXiv:math/0601130v1) takes a different route. One proves that the moduli space admits a partial compactification by adding degenerate surfaces which have nodes on the boundary. The inclusion of the moduli space in this partial compactification is a weak equivalence of orbispaces. However, the partial compactification contains a simple subspace $D$ of surfaces whose irreducible components are disks. The inclusion of this orbispace into the partial compactification is again a weak equivalence. Now note that by drawing a vertex for each disk and an edge for each nodal point we obtain a ribbon graph. This gives a orbicell decomposition of $D$ which then proves the theorem.

(3) Penner's proof is mentioned above. I am not too familiar with it, unfortunately.