I don't really have the background to understand such matters, but this presentation by Alan Guth gave me at least some idea of what it means to say the energy of a gravitational field is negative (which Guth calls a "miracle of physics").
Starting at 0:52:00 in that video clip, Guth presents this thought experiment...
...where it's taken as a given that there's no gravitational field (and hence no gravitational energy) inside a "hollow shell of mass". Guth then postulates arranging pulleys outside the shell, and having the shell shrink, so it encloses a smaller volume.
If those hypothetical pulleys were coupled to dynamos, they'd generate energy from the "collapse". So in the third diagram above, since the space between the original and final (now smaller) enclosed volume does now contain a gravitational field, and since the whole thought experiment is a "closed system" within which total energy remains constant, the (positive) energy "extracted" from the process must equal the (negative) energy of newly-present gravity in the shaded volume above.
That explanation works for me.