What does it mean, exactly, when the body is said to become more
efficient at burning fat?

Does it mean that the body burns more fat or less fat?

"Efficient" could mean "less fat" in the same way that fuel efficiency
in cars means *less* fuel used for the same distance. Thus, the more
"efficient" our bodies become at using fat, the more intense our
workouts have to be to burn the same amount as before!

But that's semantics...what's the actual physiology??

For the physiology, I think Gary Taubes' "Good Calories, Bad Calories" may
help.

The phrase seems ambiguous to me, thus, might need contextual
clarification.

I second the book recommendation.

And, yes, it is not a clear term in this context. I have seen it used to
mean that the body burns, rather than stores, a higher proportion of body
fat and hence gets more energy in the short term from the fat that is eaten.

Richard Yates