Short answer? Very carefully, and with lots of hands. :)

Of course, let's give you a little more to chew on. So there are many factors that contributed to the growth and maintenance of the Qhapaq Ñan, or Royal Road. The first of these was relatively straightforward: the Inca were not the first kids on the block. Earlier civilizations had built their own road systems in their regions: the Wari in the central highlands (including Cusco), the Tiwanaku around the southern and western portions of Lake Titicaca...more contemporary groups like the Qolla and Lupaqa in Titicaca and the Chimu on the northern coast were also building road systems to interconnect their polities. When the Inca came to an area, these existing roads were often claimed by the state, so existing routes were spruced up and widened to allow for Inca use.

The next big factor in their production and especially expansion of the road system lies with the Inca administrative behemoth. Constructing a road required labor, and lots of it. So the Inca levied their taxes through the mit'a (which I believe translates to "your turn", heh). The mit'a required able-bodied men to leave their lands to their families (often during off-season, not during the harvest or with families unable to provide the labor easily) and travel along existing roads to build administrative centers, work on Lands of the Sun and State (government fields and estates, basically), and of course, maintain and build roads and terraces. Inca architects and administrators would determine the directions of expansion, the military would protect the laborers, and the laborers would be whisked away, sometimes hundreds of miles away, to work on these projects. Storehouses (tambos) built at regular and reasonable intervals held local food, uniforms, and supplies to maintain these projects (so it's not like you needed to bring a sack lunch working for the Inca). This is how the remarkable expansions of the Qhapaq Ñan took place, with a circuit being completed around Lake Titicaca, the northern roads reaching to Quito, and especially the southwestern road that followed the Pacific coast through the driest place in the world, the pampas of Atacama.

This helps explain the maintenance of the roads during Inca times, but I wonder how much maintenance was really needed, considering the next factor: the Royal Road was only to be used for Inca business. Military and labor trains, Inca administrators, ch'aski message runners, and the Inca royal lineages (naturally) could use the roads, but all other economic or local business was done on local roads. In fact, it was illegal to use the Inca roads for non-Inca business. So they didn't experience your standard wear-and-tear, except perhaps during military campaigns.

As for the Spanish use of the roads, once the Spaniards established control over Peru, they didn't have as much need for the roads as the Incas did. Colonial motives are often not to maintain existing infrastructure of the colonized, because that polity is not inherently connected to the colonial power. New infrastructure (constructing the Spanish cities of Lima and Arequipa, for instance) redesigns the intent of a colonized region not to sustain itself but to remove commodities from the region for international trade and production, centered around the mother country. To this end, the only Inca roads and routes that were maintained to any extent were the ones that benefited this colonial effort. That desert southwestern road wasn't colonially helpful, as it was built primarily for military campaigns to reach the southern border of the empire promptly. So it was abandoned pretty quickly.