Q: How could technological advancement exist in an Anarchist state?
A: By people developing technologies.
Whereas now technological development is subsumed to the needs of the economy, funded by the government, given its first practical application in military, surveillance, etc., in anarchy the government and the capitalist system would not exist.
One could easily argue that technology is so heavily shaped by the forces of state-capital that it would not exist in anarchy. So what would happen to it? Much of it might be destroyed. Some existing technologies might be re-purposed, reappropriated, redeveloped. Some new technologies (though arguably it makes little sense to call them "technologies" if they are not formed by the state-capital) might be developed that would not have seen the light of day under this present system.
It is important to mention here that one anarchist critique of technological advancement is that it has little to nothing to do with the "benefit of people" as you put it. Largely it is geared toward the interests of government (military effectiveness, surveillance, social control, etc) and profit (hence life-saving medical care costing tens of thousands of dollars). The abolition of the state and capital, many anarchists argue, would allow technology to actually be put to people's benefit. Others argue that the abolition of technology would be to the benefit of the earth, its ecosystems, and the human species.
Finally, in response to the question of public services, some anarchists point out that all of these systems can be (and historically have been, see for example anarchists in Spain in '36) run by collectives of people managing things themselves without statist or capitalist arrangements between them. Others argue that the shape of public services is largely created by the state-capital (as are the needs for these services), so these services would be largely or entirely abolished. Instead of being publicly transported, we would travel. Instead of being given health care, we would learn healthier relationships to our bodies, including their illnesses and deaths. Instead of garbage collection we would have an ecosystemic relation to the land. Instead of libraries... well, libraries are pretty fucking cool.
(And by the way, anarchy may indeed "rule" in the vernacular sense, but "under the rule of anarchy" is a ridiculous concept.)
[edited for italics]