That's the question, isn't it? Libertarians hear it so often that it's become a cliche, a catch phrase spoken in a halfwit's voice to encapsulate our frustration with the freedom-averse. If you dare to ask it, don't be surprised if you find yourself on the receiving end of an explosion of pent-up aggravation.
But I'm not here to do that. No, I'm here to answer that question. Not every libertarian is willing to do that any more, so if you really, truly desire a sincere response, you might want to pay attention.
The obvious answer is that the same people who build roads now would build them if government stayed out of the whole business. After all, it's not the government that actually builds roads; it's construction companies, who presumably wouldn't suddenly lose their road-building expertise if Big Brother suddenly bowed out.
So, the actual question is more along the lines of, "Who will decide that roads need to be built, where they should be built and in what forms, contract with construction companies to build them, and pay for them to be built?" To which the libertarian could rightly respond, "The same sorts of people who contract with construction companies to build shopping centers, gas stations, apartment complexes, office buildings, and parking garages." In each of those cases, an entrepreneur perceives a demand for some particular type of construction, invests capital in producing it, and tries to manage it as effectively as possible to make a profit from it.
There's nothing inherently special about roads that makes this model inapplicable. Roads provide a service for which people are willing to pay, and so there's money to be made in building and managing them, just as there's money to be made with shopping malls and restaurants.
OK, so who's going to pay for this? The same people who pay for it now, of course. The major difference is that instead of the fees being hidden in gas taxes, license fees, and sales and income taxes, the costs would be transparent. How much do you pay to drive to work each day? You probably have a vague idea at best, because you're paying for it indirectly. How much money do you shell out for groceries each week? You probably have a pretty good idea, and with only a little effort you could get the exact figure, because you pay for those directly.
There would also be competition between road owners for your business, and because prices would be transparent, drivers would have a strong incentive to choose the roads that provide the best value for their money. A road that charged too much, was riddled with potholes, or prone to ugly traffic jams or deadly accidents would likely lose drivers to alternate routes, much like a restaurant with a reputation for mediocre food, high prices, poor service, and outbreaks of food poisoning would lose diners. Road owners would have a strong incentive to find the optimal balance of safety, speed, and price. It would also be to their advantage to find ways to keep traffic flowing smoothly, especially during peak hours. Traffic jams and slow commutes limit the number of cars - paying customers! - per hour.
Road owners would also find it in their interest to make their business models as painless and convenient as possible for their customers. Different roads might collect tolls, charge a subscription fee for regular drivers, or even pay for the road through billboard advertising. A business development such as a shopping mall might even maintain a stretch of the road at its own expense, and charge drivers nothing to use it. There are probably even more business models that haven't occurred to me or to you, but which some enterprising businessman or woman could dream up and put into action.
But if there are many competing road owners, wouldn't there be too many different standards? What if one uses round orange stop signs and another uses traditional red octagons? What if one has cars driving on the right side, and another on the left? Thankfully, those too would likely be non-issues, because road owners would desire that their roads be compatible with those of other road owners. Standards would quickly emerge that would allow for compatibility between the roads of different companies. A road owner who stubbornly insisted on contrary standards for no good reason would find that other owners would be reluctant to have their roads intersect with his. For the safety of their drivers - their paying customers - they would likely prefer to build overpasses or tunnel under the non-standard road, denying the stubborn road owner a lot of business. When his road goes bankrupt, another company could acquire it and run it more prudently, to the greater benefit of itself, drivers, and other road owners alike.
Competition and the profit motive would push private road owners to provide the best possible service at the lowest possible price. When they screw up, they lose customers and money. If they want to win back those customers, they need to make real improvements, and quickly. By contrast, when government roads have problems, more funding is extracted from taxpayers. Often, those funds are squandered on study after inconclusive study, with little or no improvement to actual roads. Government pays no price for dragging its feet or wasting resources, because it has a monopoly. It's going to get your money no matter how lousy a job it does. Horrible congestion or a spate of fatal accidents are cause for political grandstanding and endless argument, not decisive and efficient action.
Not only would private enterprise be more than equal to the task of building and managing roads, economics gives us every reason to believe they could do it better, more efficiently, and at lower cost than governments do it. It's simply been done by government for so long that it's frightening to many people to contemplate it any other way, and that's unfortunate. Without the freedom of the market, we never see what might be, because the people who might do it are not afforded the freedom to try. It remains forever only a vision in the minds of those who understand the engine of creativity that is the free market.