Xingyi and Drilling
My caveats will come first.
I don't believe it is possible to adequately explain the "essence" of any martial art, briefly, in print, without distortion. Even with direct access to a teacher everyday it still takes years of study, learning and practice to understand the essence of a martial art. Traditional Chinese Internal Martial Arts are very much not an exception.
Xingyiquan is a profound system. I say that first so as not to offend, because what I really mean to say is that it is profoundly simple.
Xingyiquan/Xinyiquan has been altered and changed through various branches and various lineages. I highly doubt what I say can be fully applied to each extant incarnation of the art.
Now to the "meat" of the matter,
Xingyiquan is not a very complicated art. I don't make any claims to my "mastery" of it, but I will write honestly from my understanding of it. Many people try to complicate the art though, often for either marketing purposes, their own "attachment" to it, or their own lack of wholistic understanding of the subject.
Most of the forms and training in Xingyi is an expansion upon a few common themes. First, the common themes, which you may or may not take as the "essence" of the art. (I don't believe I have the words to precisely explain the "essence" of Xingyi, much less Bagua - which I put a lot more heart into.)
1. Structure and Movement - you can also think of this as Zhan Zhuang, the half step, xingyi's "base/core" power, etc.
2. Mind/Body unity - Xing and Yi, the importance of training the mind/intent/will/spirit of and in the art.
3. Ning Zhuan - can't think of a good English word that isn't limiting (actually that's probably not even a good two Chinese "words" for it), but this includes Buddy's (likely) thought of "drilling" in all forms, clockwise and counterclockwise rotation about an axis, rise-drill-fall-overturn, expansion and contraction with rotation, the twisting of one's arms/body, etc.
And the optional #4 (which I could argue is a subset of #2 above) -
4. With maximum efficiency, control and destroy the center - the intent and strategy of Xingyi.
Xingyi, alike Bagua, follows a learning path wherein you start with nothing, get something and end with nothing and/or one thing. This is a product of the Chinese conception of a cyclic and rhythmic world view of nature.
This is easily seen in Hebei Xingyi training (but its found elsewhere as well - a mantra of Xinyi is for the ten fists to become one). I will simplify it greatly below:
From a calm, detached, centered and omni-directional potential (wuji) one progresses to santi (the essence of structure, power, center control, whole-body power, mind/body unity, and determination) then expands to the five elements (understanding of the core jins of xingyi, use of ning zhuan as it was attempted to be explained above, the base of xingyi's technique/application/strategy/tactics/...), then to a further understanding of the breadth of the possibilities inherent when one expands and changes the intent of the five elements (i.e. the animal forms, the linking forms, the other traditional xingyi long forms, the training of weapons, and attribute/conditioning training). Then the process needs to happen in reverse, wherein after one has gone from the simple to the complex, one will now go from the complex back to the simple and reintegrate what one has learned back to its "essence."
This "essence" is not a simple, explainable, logical, mental construct. It is a combination of ingrained attainment in the body, mind and spirit.