Chapter 2 of Harer's paper "The cohomology of the moduli space of curves" is good.
The point is that there is the "arc complex", a simplicial complex which gives a suitable triangulation of Teichmüller space which is compatible with the action of the mapping class group. The simplices of the simplicial complex correspond to "arc systems", which are certain collections of curves $C_i$ on an oriented surface $\Sigma$ with boundary $\partial \Sigma$ that begin and end on the boundary, and which decompose the surface into discs.
You can define a ribbon graph to be a graph together with a cyclic ordering of the edges around each vertex. To get a ribbon graph corresponding to an arc system, take the dual graph of the arc system, that is take the graph whose vertices are the components of $\Sigma \setminus (\bigcup_i C_i)$, and the edges are as you'd guess (I can't think of a nice terse way to put this into words, but it's easy to explain with a picture...), and then give the edges around each vertex a cyclic orientation ordering via the orientation of the surface.
Beware that Harer's paper doesn't mention ribbon graphs, nor do some of the other standard references. This had confused me for a while (an embarrassingly long time, in fact) when I was trying to read about this in the literature. But as I said, just know that the ribbon graph picture is just the dual picture to the arc system picture.
P.S. Some more references here.