It's the stuff on which society rests. The Wikipedia article notes that it can be "owned and operated by government or private companies" and reveals that it's a term quite hard to define; it includes not just roads and bridges, but perhaps even education and health systems. In fact, I'd say that pretty well any large industry (whether made up of many small firms or a small number of large ones) is essential for society to operate, and therefore could be included in this rather woolly term, "infrastructure."
One way to think about it: take it away, and society would be in big trouble. So if government operates a part of the infrastructure, when government evaporates will society collapse?
No way. But I sense an obligation to show why.
In my Vision of Liberty there are chapters on how various aspects of life will probably take shape, in the coming Zero Government Society; ownership, health, education, work, wealth, justice, morals, money, travel, leisure, and religion. Some of those could be classified as "infrastructure", depending on how the term is defined. It happens not to include ideas about electric power grids, nor much about roads and bridges, water supply or sewers, etc, so perhaps this ZGBlog will make up some of the shortfall.
It's worth noting first that government did not precede the construction of those important subsystems. Humans found and drank water (and possibly washed) for tens of thousands of years before the dolorous invention of government. In America, roads were built for profit (often called "turnpikes" for one mechanism of toll collection) before government began to throw tax money at them whether or not they were in economic demand. Railroad construction did not at all depend on government (except for the need of a waiver of its ludicrous claim to "own" all undeveloped land) and notably one (the Great Northern Railway) was built soundly and profitably without using a red cent of tax money.
Government's giant leap into the funding and management of infrastructure came with Abraham Lincoln, who worked for it for all of his career prior to the War to Prevent Secession, as detailed in Thomas DiLorenzo's brilliant The Real Lincoln. Crony capitalism was in its prime; railroad builders lobbied tirelessly for tax money to provide the capital they needed. It was so very much easier to raise, than to float shares and persuade investors! - and having obtained it, they were far more careless with the funds than they would have to have been if answering to real shareholders.
So infrastructure need not be associated with government; only political pressure makes it so. Railroads, Interstates, air traffic control, gas pipelines, hydro electric plants and all other high-cost projects can and should have been run on the simple and adequate basis of free-market investment for gain. They may be formidably expensive, but no matter; they will cost less and be more efficiently run when real people have their own money at stake than when government confiscates that money and has a committee of bureaucrats controlling its use.
In the coming ZGS that will still be true, and will apply in every case because taxation will no longer exist. There will in addition be a factor operating vigorously that is absent today: competition. Is a national grid linking power stations together the best way to make electricity available? When a minor storm knocks trees down on to power lines and darkens a neighborhood, one has time to reflect on that question. Nuclear power plants small enough to place in a submarine have been around for half a century; might they not be adapted for use by the owner of an apartment building, or a small group of houses? When tax has been removed from the price of hydrocarbons, might not a simple motor generator in the basement supply electricity for a residence? And will not the market develop even safer, greener alternatives, once it has the chance? Of course it will.
The Recovery of Stolen Roads gives some account of how a ZGS will probably arrange the building and maintenance of roads, and Dr Walter Block's book The Privatization of Roads and Highways has more - so I'll omit that important subject here.
Free-market water supply and sewage removal is easy to visualize where population density is low, for often today, property owners take care of both with an artesian well and a septic system - without any government participation. In towns and cities it's less easy, for typically office and apartment users don't control any land where those might be dug. So governments supply both, using pipes and treatment plants. The maintenance of both systems becomes therefore a political act, so water of poor quality or inadequate quantity may result, as recently in Flint, MI and the whole of CA. Come the ZGS, those systems will be adopted by commercial enterprises, and will operate to maximize profit, by optimizing customer satisfaction. The price will be affordable, remember, because municipal taxes will no longer be levied. Drinking water, meanwhile, has already for many years been a staple item on supermarket shelves, usually at a price exceeding that of gasoline.
So when a demand arises, the market responds. The incentives of profit and competition will drive the free-market alternative in the ZGS, and the form they will take is hard to predict; as a city government folds, for example, little time will pass before the cry arises for water and drains so there will be ample demand to encourage entrepreneurs to take over those systems and run them for profit. By "take over" I mean that rival firms will bid to acquire the plant and equipment, not by buying them from the municipality because that never owned them in the first place and will no longer exist anyway. Rather, the first on the scene will probably register his title claim on that abandoned property and then auction it - rather as smart geeks did in the early days of the Internet; they registered potentially desirable URLs, then sold them to the large firms that could ill afford to do without them but were a little late to the party. (That applied to me; when launching the Freedom Academy in 2006 I wanted to use TOLFA.com, but found it was already reserved for a small town in Italy. I didn't offer to buy it from them, but settled instead for TOLFA.us.)
The bottom line is that infrastructure is no different from any other good or service, and will be furnished wherever there is a demand (the wish, plus the money) by anyone able and willing to earn a profit by doing so. That's true of physical facilities like those above, and for intangible ones like the industries of justice, health, education and security. The whole argument that infrastructure is a special case that only government can handle is bogus from tip to toe.