It's security, physical and emotional comfort and a sense that the world is full of opportunities, rather than discomfort, insecurity and restriction.
Of course, there are children in all socio economic groups who experience adverse experiences such as illness, SEND, bereavement, abuse or neglect, but the confident, poised and secure ones that are described as having Public School Energy are the ones who have not known a world where those things are absent or could be swept away in an instant. They don't even need to be consciously aware that their needs are going to met - they just are. They know how to say 'big' words because they've heard them used and they're just regular ones - nobody's going to laugh at them for mispronouncing a word they've only ever read or getting ideas above their station for saying 'Oh, absolutely' rather than 'Yeh'. Which means that the possibilities are endless and nobody in their experience is going to 'put them straight' (ie, ask them who the hell they think they are and what on Earth possessed them to imagine that they are worthy of such things) or deliberately knock them down because they're not worthy of such high-faluting notions.
Even something like not doing their homework is down to their choices and actions, rather than not having the equipment, somewhere suitable to do it, being too cold, no internet, a parent that doesn't have the literacy, numeracy or interest in algebra/learning as a whole to facilitate or help with it - it's something that they have power over, which means any punishment for not doing it is a consequence of their actions, not from external circumstances they have no control over. So they have a sense of agency.
It's a life that everybody should have but only a few do. I don't begrudge it them for an instant, but it is easy for it to rankle somewhat when they give away their intrinsic sense of comfort through not understanding this is not the norm when dealing with other people. In their own bubble, there's no reason for this to matter and probably goes some way in explaining how those in the Public Eye can be so distressed/angry when they feel they're being criticised or attacked or things begin to go wrong. They've not had the same experiences - their entire existence is predicated upon their world being a good place.
That's Public School Energy. The world is safe, comfortable and there for them to experience and explore in the knowledge that whatever may happen, it'll all turn out alright in the end. Because that's how it's always been for them and everyone around them.