In Marxist terms, the state is a tool of class oppression, it is the machinery by which one class imposes its will on all others. Under capitalism (or rather, under a dictatorship of the bourgeoisie (DotB), like e.g. the US, UK, France, Brazil, etc.) it is the bourgeois class who controls the state and uses it to oppress the proletariat (and any remnants of prior classes like the peasantry). In a dictatorship of the proletariat (DotP), that structure is inverted and the proletariat uses the state to oppress the bourgeoisie (and any remnants of say, aristocracy or whatever).

The idea is that in a DotP, the bourgeoisie will eventually become proletarianized and, after a long enough period when there are no more bourgeoisie, there will be no more proletariat (as classes are defined in terms of the conflict between them). Without a class system, there will be no longer any reason for the state - as that tool of class oppression to exist. All that will be left in terms of governance, will be the administration of things. You still need to manage healthcare, housing, transport, etc.