I think Nagel has no clue what "Principle of Sufficient Reason" really means. Perhaps he's right that analytic philosophers have traditionally felt comfortable with assuming ultimate, brute facts which are intelligible without any further context, but this has everything to do with nominalism (in the sense in which D. Armstrong accused W.V. Quine of Ostrich Nominalism) or anti-transcendentalism (in the sense in which Foucault, Kuhn and Hacking are transcendentalists - because they think any fact is intelligible only in a broader context of sense-making), although I wouldn't be so sure (ex. Wittgenstein, both early and late), and nothing with the Principle of Sufficient Reason.
The Principle of Sufficient Reason is what we would nowadays call: The Law of Causality. It says that, for any X that happened, there is such Y whose happening explains X's happening, which means that a) Y happened and b) had Y not happened, X wouldn't either.
A good example of the use of the Principle of Sufficient Reason is how Leibniz justifies his Principle of Identity of Indiscernibles by invoking God. If two indiscernible objects were nevertheless different, God would've acted without any reason by distinguishing them (instantiating one and not the other). But God necessarily acts rationally, therefore: the Principle of Identity of Indiscernibles holds. He uses similar arguments against Newtonian absolute space (and non-relationist theories of space generally), cf. David J. Hyder, Three Paradoxces Concerning Causality and Time: Parmenides, Leibniz, Einstein/Schrödinger.