This was a really inspiring speech. The entire speech is very valuable to read, but to give a TLDR: the new aristocracy is defined as the elite that gains their wealth through their labour rather than their assets. It's new because 50 years ago, the elite was more based on capital than their own work. It points out that you can exploit your capital without damage to yourself, but you can't exploit your own capacity to work as then you use yourself up, but that's precisely the trend as work amount expectations for elite lawyers have doubled in recent decades. The consequence of this situation is pointed out this way:
> To live in this way is, quite literally, to use oneself up. Such a life proceeds under a pervasive shadow: at its worst, it squanders the capacity to set and pursue authentically embraced, intrinsically valued, goals; even at its best, this life invites deep alienation.
With alienation they probably mean something like not seeing your parents, friends, family, etc for years. The proposed solution to this dilemma is then presented in the key paragraph:
> The new aristocracy promotes human flourishing for no one: certainly not for the excluded rest; nor even for the ensnared rich. We are trained to think of economic inequality as presenting a zero sum game: to suppose that redistribution to benefit the bottom must burden the top. But this is not such a case: reforms that democratize training and talent would benefit everybody. Such democratic reforms would restore the bulk of Americans to full participation in an economic and social order from which they have been, for several decades now, increasingly excluded. And democratic reforms invite the elite—you all—to accept an almost costless diminution in wealth and status in exchange for a massive, precious increase in leisure and liberty, a reclaiming of your authentic selves.