O(mLog^2(n)) algorithm for Symmetric Diagonally Dominant Linear system: How can I understand it?

Some links to background and relevant papers appear on this webpage from Dan Spielman.

I'm not familiar with these techniques either, but it seems there are a few steps to understand the significance of this.

1. An important class of linear systems is the symmetric diagonal dominant systems, where the A matrix (the system is Ax = b) is symmetric and has A_ii >= sum(j != i, A_ij) (that is, the diagonal is larger than the sum of the off-diagonal elements).
2. A weighted undirected graph can be caracterized by its laplacian matrix L, which is, given an arbitrary indexing of its vertices, L_(i,j) = -w_ij (where w_ij is the weight of the edge between vertices i and j) and L_(i,i) = sum(i != j, w_ij).
3. Since a graph (and an enumeration) uniquely specifies its laplacian (and vice versa, as you can easily check), you can actually talk of any laplacian-like matrix as a graph.
4. You can talk of something called a graph preconditioner, which is a graph G' that approximates a given graph G but is easier to solve (i.e., invert the laplacian).
5. There is this concept called stretch of a tree approximation to a given graph G. The stretch of an edge e connecting vertices i and j is the ratio between the sum of the inverted weights of the edges the (only) path in the tree connecting i and j and the inverted weight of the ij edge in the graph. sum(w'_ab for ab in the path)/w'_ij
6. You can then talk about an alpha-sparsifier, which is a graph G' that has a nearly linear number of edges and alpha-approximates a given graph G in terms of something related to stretch.

This paper, then, develops what they call an incremental sparsifier, which is a graph that approximates another given graph as you add edges in a specific way. With this guaranteed approximation they can build a solver that runs in the alloted time.

I've just summarized their main claims here, and I have in no way verified or undersood their results.