cosmology - what is the current explanation for the formation of cosmic voids?

You might like to watch this short movie from one of the authorities on this subject:
https://www.youtube.com/watch?v=wI12X2zczqI

The movie starts by answering the question "what is the cosmic web and what does it look like?". Simulations can reproduce such structures, but they do not explain "why" they look the way they do. The clue lies in the geometric appearance of the structures: The voids resemble the elements of a Voronoi tessellation in which the walls of the Voronoi polyhedra intersect in lines that are identified with the filaments. We need to understand the dynamics that generated this pattern.

At around 2:30 the movie addresses the issue of why it looks the way it does using a simple ballistic model for the motion of a set of discrete particles that represent a random density field (the Zel'dovich model). The filaments are there but too fuzzy. So at ~3:00 it generalizes this so as to achieve greater realism by making the particles sticky. This is the "adhesion model", which is governed by the Burgers-Hopf equation, which is easily solved by geometric means. There is no gravity in this - it's ballistics acting on density field generated by a Gaussian Random process with a known covariance function (or power spectrum).

The adhesion model is a remarkable representation of the structure. It shows how nothing "special" is required to generate the pattern of the cosmic web. That's the "how".

The "why does the structure look like it does?" is a little more complex: this is the question of emergent geometry which is understood in terms of the technical language of the Lagrangian description of the flow. But the graphics is pretty and quite didactic, so worth a look. This happens after around 5:30.

The movie compares appearance of the flow simultaneously in both Lagrangian and Eulerian space. In the Lagrangian picture the particles have fixed coordinates,

A short descriptive article goes with the movie:
http://arxiv.org/pdf/1205.1669v1.pdf
and a short explanatin of the Lagrangiuan view is at
http://arxiv.org/pdf/1211.5385v1.pdf
This is part of the thesis work of Johan Hidding from the Kapteyn Instutue in Groningen (supevisor Rien van de Weygaert - the father of the cosmic web according to Richard Bond). A full presentation of the nature of the singularities in terms of Morse Theory is at
http://arxiv.org/pdf/1311.7134v1.pdf
and there are several other papers pending.