Hoagland discovered that where the points of a tetrahedron touch the earth's surface from the inside, energy is coming in from "somewhere" and it's not this dimension, so it must be hyperdimensional. But after this basic observation, his theory gets a bit vague, more like he has clues pointing toward something strange going on having to do with geometry and quantum physics, but he doesn't know what exactly is going on or how apply it other than through observation and guesses.
You can stick other shapes inside the earth, like a cube, dodecahedron, octohedron, or icosahedron and find other significant points on the surface. Hoagland focuses heavily on the tetrahedral model despite these other shapes being just as important.
What's going on at these points? Well, the vertices of platonic solids show where standing waves collect on a sphere. If you vibrate a sphere, you'll get cymatic waveshapes with nodes and antinodes arranged geometrically on the surface.
Whatever energy vibrates from the center of the earth, it seems to create standing waves on the surface. Perhaps it's the nodes, perhaps the antinodes, but at these points strange things happen. They are major earth grid points, vortices, portals so to speak. Place the proper device in that region and you can gather energy or even control the rest of the grid to an extent (that idea was explored in Umberto Eco's book Foucault's Pendulum, if you've read it).
But where these points are located depends on how you orient the platonic solid inside the earth. Here's a map showing latitudes where the gridpoints would be if the platonic shapes were aligned with the north geographic pole:
You can see the tetrahedral one going through Hawaii. The other lines pass through such places as the HAARP facility, geomagnetic hotspots on the globe, the various secret bases in Australia, Canary Islands, and so on.
As to how a pyramid works, the only explanations I read that make sense come from the Ra and Cass material. The first explained that pyramids act as funnels or diodes for etheric energy that is always present around us, causing a one-way flow. The C's then expounded upon this, saying that when put at one of these grid points, the "magnetic energy streams" coming into and out of that point, streams which are normally paired in mutually balancing flows of opposite direction, become unbalanced. Nature then expends energy to try and restore equillbrium, but this energy can be collected for as long as the unbalancer (the pyramids) remain actively in place.
I'm not sure how a device can be built to capture this energy, but there are enough annecdotal reports of fringe inventions to suggest it's real, possible, and has been done.
Acquiring fringe knowledge is like digging for diamonds in a mine field.