First, a few reading resources:
http://www.ejnet.org/landfills/
and
http://dnr.mo.gov/bridgeton/
and
https://en.wikipedia.org/wiki/West_Lake_Landfill
Now, the executive summary:
The Bridgeton landfill was originally a limestone quarry, which means that there were never any caves or tunnels at the site; just a really huge hole in the ground. They laid some pipes for collecting any nasty chemicals that could leach out over time, then sealed off the sides and bottom with clay and/or plastic, and proceeded to dump trash into this great big hole in the ground until the trash was more than than 150 feet deep. Whenever any section of a landfill was filled up, that section was "capped off" in much the same way that the bottom and sides of the landfill were sealed, and then they moved on to the next section.
Over time, the particular mixture of waste that gets dumped into a landfill begins to decompose and liquefy and undergo a whole host of chemical reactions to produce some of the nastiest substances out there. In general, this stuff likes to burn but doesn't have enough oxygen to do so; dirt is slightly permeable to oxygen, however, and bulk waste even more so. As such, once any part of a landfill has caught fire, the area underneath will burn and spread underground in the same way that charcoal does once you've smothered the fire itself: it simply glows and oxidizes and therefore burns extremely slowly without an open flame.
Subsurface fires are almost impossible to put out, as demonstrated by the case of the
Centralia coal mine fire that has been burning for more than 50 years; even once all of the surface fires were doused and all of the mine tunnels were sealed off to prevent oxygen from feeding the flames, the coal continued to smolder and burn underground. The only feasible way to deal with an underground fire is to contain it as best as possible and let it burn itself out; in the case of the Centralia fire, veins of coal stretch out and down through soil and stone in all directions, providing far more fuel for the fire than could ever have been excavated by mine workers even if the mine hadn't been closed off. In the case of the Bridgeton landfill, the entire thing is essentially no different from single a giant lump of coal burning slowly from the inside out; so far, all efforts to prevent the spread of the "fire" under the "ground" have been ineffective.