Explicit homeomorphism between Thurston's compactification of Teichmuller space and the closed disc Thurston's celebrated compactification of Teichmuller space was first described in his famous Bulletin paper. Teichmuller space is famously homeomorphic to an open disc of some dimension (this can be seen using Fenchel-Nielsen coordinates), which is $6g-6$ for closed surfaces of genus $g\geq 2$.
Thurston embeds Teichmuller space in some infinite-dimensional natural space (the projective space of all real functions on isotopy classes of simple closed curves) and studies its closure there. The closure is realised by adding some points that correspond geometrically to some particular objects (called projective measured foliations). The added points are homeomorphic to a sphere of dimension $6g-7$ and the resulting topological space is just a closed disc, the new points forming its boundary.
The Bulletin paper contains almost no proofs. The only complete proofs I know for this beautiful piece of mathematics is described in the book Travaux de Thurston sur les surfaces of Fathi-Laudenbach-Ponearu (an english translation written by Kim and Margalit is available here). The homeomorphism between the space of projective measured laminations and the sphere $S^{6g-7}$ as explained there is clear and natural, it's obtained by re-adapting the Fenchel-Nielsen coordinates to the context of measured foliations.
The proof that the whole compactified space is homeomorphic to $D^{6g-6}$ is however more involved and less direct. First they study some charts to prove that we get a topological manifold with boundary, and that's ok. The compactification is thus a topological manifold with boundary homeomorphic to $S^{6g-7}$, whose interior is homeomorphic to an open ball of dimension $6g-6$. Are we done to conclude that the compactification is a closed disc? Yes, but only by invoking a couple of deep results: the existence of a collar for topological manifolds, and the topological Schoenﬂies Theorem in high dimension. That's the argument used in the book.

Is there a more direct description of the homeomorphism between Thurston's compactification and the closed disc $D^{6g-6}$?

Is there in particular a Fenchel-Nielsen-like parametrization of the whole compactification?
 A: One natural attempt to compactify Teichmuller space is by the visual sphere of the Teichuller metric. However, Anna Lenzhen showed that there are Teichmuller geodesics which do not limit to $PMF$ (in fact, I think it was known before by Kerckhoff that the visual compactification is not Thurston's compactification).
However, it was shown by Cormac Walsh that if one takes Thurston's Lipschitz (asymmetric) metric on Teichmuller space, and take the horofunction compactification of this metric, one gets Thurston's compactification of Teichmuller space. In fact, he shows in Corollary 1.1 that every geodesic in the Lipschitz metric converges in the forward direction to a point in Thurston's boundary. I think this gives a new proof that Thurston's compactification gives a ball.
As Misha points out, it's not clear that the horofunction compactification is a ball.
Another approach was given by Mike Wolf, who gave a compactification in terms of harmonic maps, in The Teichmüller theory of harmonic maps, and showed that this is equivalent to Thurston's compactification (Theorem 4.1 of the paper). Wolf shows that given a Riemann surface $\sigma \in \mathcal{T}_g$, there is a unique harmonic map to any other Riemann surface $\rho \in \mathcal{T}_g$ which has an associated quadratic differential $\Phi(\sigma,\rho) dz^2 \in QD(\sigma)$ ($QD(\sigma)$ is naturally a linear space homeomorphic to $\mathbb{R}^{6g-6}$). Wolf shows that this is a continuous bijection between $\mathcal{T}_g$ and $QD(\sigma)$, and shows that the compactification of $QD(\sigma)$ by rays is homeomorphic to Thurston's compactification $\overline{\mathcal{T}_g}$ in Theorem 4.1. I skimmed through the proof, and as far as I can tell the proof of the homeomorphism does not appeal to the fact that Thurston's compactification is a ball, so I think this might give another proof that it is a ball.
A: I am not aware of a single Fenchel-Nielsen type parameterization as you ask for, and I'm not sure there can be one, because even on the boundary sphere the manner in which Fenchel-Nielsen coordinates are "re-adapted" does not produce a single coordinate system. The way that proof gets coordinates for the boundary sphere does start with a pants decomposition of the surface, as do Fenchel-Nielsen coordinates. And one does get a single coordinate system for the open subset of the boundary sphere which has nontrivial intersection number with each pants curve. But then one has to patch in additional coordinate charts to cover the closed subset of measure foliations that have zero intersection number with one or more pants curve. The proof does demonstrate that these coordinates can be patched together in such an explicit way that one can see the homeomorphism to a sphere, but nonetheless one is still patching things up.
Edit: I also recall that in one of his very earliest writings on this topic, Thurston gave a different proof, certainly not explicit, that the boundary is a sphere. Namely, from the existence of a pseudo-Anosov homeomorphism $\phi$, which acts with attractor--repeller dynamics, one gets a covering by two open sets homeomorphic to Euclidean space: for any neighborhood $U_+$ of the attracting fixed point and any neighborhood $U_-$ of the repelling fixed point there exists $n>0$ such that $\phi^n(U_-)$ and $U_+$ cover the boundary. It follows that the boundary is homeomorphic to a sphere. I posted this question to verify that the same was true for manifolds with boundary, and so the same proof works for compactified Teichmuller space, as Thurston undoubtedly knew: the action of a pseudo-Anosov homeomorphism on compactified Teichmuller space also has attractor-repeller dynamics, and so it is covered by two manifold-with-boundary coordinate charts, and so it is homeomorphic to a closed ball.
