They didn't conquer an empire of millions. They conquered the elites. As best as we can tell, from mexico to parts of south america, they expertly pitted groups of native elites against each other and somehow ended up on top. Nobody really knows how or why they were able to succeed. But even more impressive than the conquest is their staying power. The spanish essentially took over the elites and managed to maintain that position for hundreds of years.

Take mexico for example. After the spaniards positioned themselves atop mexican society 400+ years ago, they never relinquished that position. Even though ethnic spaniards made up anywhere from 1% to 5% of total mexican population thoughout the 400 years. Even more astounding is that the spaniards used their position atop mexican society to outbreed the native population. 66% of current mexican Y chromosomes are of european ancestry, while 90% of mexican mitochondrial dna comes from native ancestry. Fascinating stuff.

The spaniard conquest of central and south america differed markedly from the genocidal conquest of north america which is reflected in the genetic makeup of the populations in the americas.