I'll take a crack at this one. I've been thinking about the same question lately, and I think I've reached a satisfactory answer. Whether I can articulate that answer briefly with any clarity whatsoever is another matter... Let's see.
First, ressentiment is basically the process by which the slave psychologically overcomes his bondage by thinking of himself as the moral better of his master. ("Yeah, you may have me under your boot, but so what -- _you_ are a totally despicable human being!"). What this leads to are the adoption of Christian values (weakness, submission, hatred of the body) that run against healthy, natural "aristocratic" virtues (strength, domination, love of the body).
N. seems to have had a lot of admiration for caste-based societies where everyone knows his proper role and that was the end of it. Much of his hatred for socialists and anarchists seems to revolve around how they're riling the groundlings up and taking them out of their happy slumber: socialism and anarchism were really very Christian in that they promoted hatred of the ruling classes simply because they _ruled_.
(If you listened to Doug's great interview with Ian Bone a few weeks ago, what was particularly striking to me was his insistence that we ought to hate the rich, not just in the abstract, but _personally_ as individuals -- there is a deeply moral component to this sentiment -- I remember thinking "ah ha! this is what N. was writing about when he was talking shit about anarchists").
Okay, how can this view at all be reconciled with an egalitarian, anti-capitalist political project? I think the answer lies in that the left seems to be divided between those who view politics as a _moral_ struggle versus those who view politics as a _power_ struggle: The moral folks see their own moral superiority as their primary advantage, and they're driven by essentially moral considerations: they're the ones who want "justice" for workers, minorities, women, etc.. Compare this stance with that of the power folks, for whom "justice" is irrelevant -- they'll make their own justice damnit: they don't need a moral imperative or any other ethereal justification to seize the means of production -- they'll do it because they _want_ to. Fuck "justice"! For the power folks, politics needn't be personal, your adversaries don't have to be rotten people -- its okay to respect your adversaries -- you just have to beat them.
So I guess N.'s notion of ressentiment is helpful to those of us on the left who view politics as a power struggle; it explains the deficiencies of having morality sit at the center of one's political engagement.
-WD