How were a handful of Spaniards able to subdue the populous and powerful Aztecs and Inca empires?

/u/Mictlantecuhtli has provided some excellent links on this subject, but I'd like to just briefly touch on a few points.

1. A handful of Spanish didn't subdue the Aztecs

During the war against the Mexica there were about 1000 Spanish in Mesoamerica (with resupply and reinforcements coming from Cuba to replace lost equipment and men). Fighting right alongside that handful, however, were (by Cortés' own estimate) about 100,000 Tlaxcalans, Acolhua, Totonacs, and others. Even then...

2. The Battle and Siege of Tenochtitlan was not easy

The actual fighting part of the Conquest itself took the better part of a year. The above noted confederation of forces entered the Valley of Mexico shortly after Christmas 1520, and it was not until mid-August that Cuauhtemoc surrendered. Between those dates was a grueling and grinding 3 month siege which itself was preceded by months of intense fighting around the lakes and on the causeways into Tenochtitlan. So it's important to keep in mind that this was no cakewalk. Perhaps more important to keep in mind is that...

3. Framing the Conquest as a clash of cultures is anachronistic

In retrospect, what happened in 1520-1521 Mexico was a world changing event with far reaching repercussions. At the time though, it was not framed in such stark terms, because the conceptualization of an American/European dichotomy did not yet exist. The notion of "primitive" peoples being "naturally" subdued and subsumed by an "advanced" race is something that took centuries to develop. Cañizares-Esquerra's How to Write the History of the New World: Histories, Epistimologies, and Indentities in the Eighteenth-Century Atlantic World is (as the subtitle hints) perhaps a bit overly academic for a quick overview, but part of the thrust of the book is that the early Spanish essentially viewed the Aztecs (and other groups) on equal footing. They were vile pagans, true, but they were not innate inferior. Later (primarily French and English) writers, far removed in both space and time from the events of Contact, would discount early Spanish texts as both gullible and fanciful, while also viewing native sources as proof of their inherent inferiority. The point being that neither side was approaching the Battle of Tenochtitlan as a clash of cultures. Also consider...

4. It took time for things to change, if at all

The process of synthesis and assimilation of cultures in Mexico was a decades (if not centuries or even on-going) process. Immediately post-Conquest, the ultimate head of state may have shifted from Tenochtitlan to Madrid, but the Mexica who had been happily conquering their way around Mesoamerica, went right on conquering their way around Mesoamerica. The problem of "invisible Indians" is particular severe when discussing post-Conquest expeditions to the North and South of Central Mexico. Every one of them was ostensibly a Spanish led mission, but native elites served right along side and indigenous troops often made up the bulk of the soldiers. Ironically, the Conquest led to opportunities for a Nahua identity to further spread and cement itself in Mesoamerica.

I've written on these topics several times, so here are some past comments you might find interesting:

I'm more than happy to field any follow-up questions you might have.