Reading some comments and answers on here gave me another idea about how they do it -- and explains why glass ant colonies taken into captivity never seem to recreate their unique glass tunnels even in ample-sized sandboxes (pardon the pun): Lightning rods.
Finding a place to call home
When a new queen seeks to build a colony, she takes a couple hundred (maybe a thousand or two?) worker ants with her. They leave the nest at the first sign of precipitation; this has a threefold benefit:
The coming rains will ensure the queen and her "borrowed" workers stay hydrated while they seek an adequate location for a nest.
Rainfall is frequently accompanied by lightning storms... and this is how they start their new nest.
Once the nest has been formed, rainfall collects in reservoirs at the bottom of the nest to provide a source of moisture as well as maintain ambient humidity in the nest.
As the first raindrops begin to fall, the ants use the flow of water to help find high ground (which also helps avoid drowning while they're excavating their tunnels). Once an ideal spot is found, some workers begin digging in the sand, periodically applying a thin layer of resinous secretions to help keep it from collapsing as they dig. While this resin isn't enough by itself to fully seal the tunnel for long term use, it adds structural enhancement and provides a key characteristic for the "solidification" of the tunnels...
A conducive atmosphere
Once they have the tunnels in place, the workers begin climbing atop one another at the entrance, forming a tower; these towers are estimated to reach nearly three meters tall* and are easily the tallest objects in the nearby landscape. Having been freshly hydrated, their bodies are rather conductive, and thus "attract" lightning. The trick is that the resinous secretions are also conductive, so once the lightning strikes the "ant rod" it carries through the secretions along the tunnel walls and fuses the silica into glass, taking advantage of a natural phenomena to help strengthen and seal their home.
Of course the ants that form the ant rod are killed in the process, but this provides a handy source of food for the remaining ants. It may not be the most nutritious, but it buys time for the colony to establish itself. The remaining ants will carry the burnt remains into the tunnels, and also scout them for any gaps that may not have fused. More resin secretions will be used to fill the gaps, and once cured is nearly as transparent and nonporous as the glass, while lending additional strength to the structure.
Similar to the Cephalotes genera of ants pointed out by @LoneBoat, the colony has a caste of large-headed workers whose sole purpose is to block the entrance(s) to the tunnel. In addition to providing defense and sealing in moisture, the "blockers" also serve to regulate water intake during desert rains. Most of the surface area of their head is a nonporous, chitinous material that serves as armor as well as a moisture barrier; most blockers have two to three small dimples in this barrier, and each dimple contains a tiny sensory receptor in its center about 100-200µm in diameter.
This receptor lets them detect moisture, and they will unblock the hole when rains come so water can come in to replenish the reservoirs. As the water pours in, regular workers scurry down to the reservoirs to monitor them; once the reservoirs are nearly full, the workers make for the tunnel entrances and gently nip the blockers who will promptly return to their position and seal off the entrances so the tunnels don't flood.
Once back in position, the blockers will refuse to move until their receptors tell them it's dry outside. While the colony will attempt to locate on high ground (improving probability of lightning strikes and minimizing flood risk) localized flooding can still pose a threat, so the workers will remain in place keeping the tunnels sealed.
* Individual worker ants are about 15-20mm in length, and the expedition's initial estimates suggest that the towers are comprised of 500-800 ants and the height and diameter is roughly relational to how many total ants the new queen was able to recruit. Naturally, more recruits means a taller and stronger tower, which improves the tower's chances of successfully getting hit by lightning to form their new home and thus improves the colony's chances of survival.