The way it worked in Spain during the Middle Ages it often was the highest powers who looked at the map, saw a habitable spot in an empty area and either sent a bunch of people to start building there (this was the case of Bilbao, for example) or delimited an area, picked a spot to be “the village” and set up a series of “startup laws” (this was the case for many of the towns which are now on the French Camino de Santiago - the kings did this during the early part of the Reconquista because they wanted to move commerce and travel inland). There were other cases where a town grew up spontaneously at a point where TPTB had created an important logistic knot: there are several towns called Puente (de) la Reina, Queensbridge, in spots where a Queen ordered a bridge built midway between two not-so-close fords and then villages sprouted up.
During Franco’s regime there were several so-called villas de nueva planta, small towns which were started up in low-population areas to exploit local resources. Some were created by the regime, some by corporations, but in any case they’d build the exploitation and the housing for the workers, and move the workers in. A few of these have ended up abandoned (too much “in the middle of nowhere”); others have grown, prospered and diversified their sources of income, but they remain small by Spanish standards, while many of the Middle Ages foundations are now large towns or even cities (Burgos, Bilbao).
Currently, the most common trigger is the need for suburbia, triggered in part by the fashion we’ve had in the last couple decades for individual houses over flats: for example Ribas Vacíamadrid, whose second word means “to empty Madrid out” - if that isn’t meant to be a “bedroom village”, I don’t know what is!
In the US, then, it works this way. A developer with a big parcel of land senses a market opportunity, lays out and plats/subdivides the property, and sells it to purchasers. The developer may build many of the houses or buildings itself, or it may just be a master developer selling to merchant builders. When the population reaches a critical mass, if local politics have not already decided the question, the residents may seek to incorporate as an independent municipality. In other cases, the new development is annexed to a nearby town or remains unincorporated.
Who funds the infrastructure? Well, that can get pretty complex, but generally the developer will try to bank against the future value of the land, using a combination of construction loans, tax-increment financing, and special taxing districts, so that the future residents will pay off the cost of their roads and pipes. States and counties (even municipalities) anxious to have the future property and sales taxes may kick in some money, or school and road improvements. For even the best-funded developers, the process becomes a race between carrying costs and income from sales. If a recession interrupts, the enterprise often fails, and new towns that were successful for the original developer are not that common. I don’t know what part of the US you’re in, but chances are there’s a project near you that was begun as a “new town” dream but ended up just another subdivision.
From the way you ask the question, I think it’s vital to note that this is all undertaken by the private market (though often having rather cozy relationships with local officials). Twice the US government has tried to jump-start the process, first with the three Greenbelt towns begun in the 1930s before the program was ruled unconstitutional, and then with the HUD New Towns begun from 1968 to 1975, which ran into the oil crisis and bad recession of 1974.