The state is a tool wielded by one class, the ruling class, to oppress an other, to keep the other class away from power and from means to take power. Under socialism, it's the proletariat who is the ruling class, it's the dictatorship of the proletariat. The remaining bourgeois after the revolution are stripped of their private properties and political power, maybe even of their personal wealth if need be.

In this state of affaire, the bourgeoisie canot continue their accumulation of wealth and are left only with what they had already accumulated, if it was not confiscated, being unable to continue to extract surplus value, their only choice is to either live of whatever wealth they have left without working or start working.

Either way, in this situation, the bourgeoisie as a class start slowly but surely vanishing, at the start the bitter dispossessed bourgeois will try everything in their power to get back their power and privileges, but as time passes, as the proletarianized bourgeois and their descendant get used to live as worker in a proletarian society, as the schools teach proletarian values, as the culture is cleansed of it's bourgeois influence and as standards of living start rising uniformly, the "bourgeois mentality" will die out on it's own. "We already live well, every necessity is free and everything we might want is affordable, why should we risk so much for the faint hope of getting back those untold riches you say you used to possess?" Their children will say.

And by this process, the bourgeois will one by one become proletarians until the bourgeois class finally disappear, and with it, classes as a whole will disappear.

Like said at the beginning, the point of the state is to oppress one class on behalf of an other, therefore, with the disparition of classes, the state stop having a purpose, and without purpose, all it's means of oppression (military, police, prisons, etc...), now useless, are dropped, and like this, the state dies, leaving in it's wake a simple administrative body without any means of oppression.

This is what I understand about the withering of the state based on my still very limited understanding of theory, if someone more knowledgeable sees this, please let me know what you think of my explanation ^^.