How did the rise of absolute monarchs contribute to the discovery of America?
An interesting question. It is not completely accurate to suggest that absolute monarchy led to the discovery of America. It would perhaps be better to say that the discovery of the New World helped bolster and empower absolute monarchy in some cases, namely in Spain and France.
Spain was ruled by an absolute monarchy during the 1400s and 1500s, the peak of the Age of Exploration; this meant the country had wealth enough to finance exploration, and subsequent wealth from the Americas was fed back to the monarchy. However, the Spanish Armada was defeated by the English in 1588, which marked the beginning of their decline.
The English, notably, were not ruled by an absolute monarchy during this time. They were able to establish colonies in the New World under Elizabeth I, which helped increase their wealth through trade.
France’s absolute monarchy was bolstered by New World trade during the fifteenth and sixteenth centuries, increased wealth helping to calm some of the aggression that had arisen against the kings.
We cannot absolutely correlate the political system of rabsolute monarchy with discovery, exploration, and colonization. Athens, an ancient Greek democracy, and Republican Rome both planted far-flung colonies. The first Europeans to discover North Amnerica were actually the Vikings, who did not have centralized monarchies, but instead looser confederations of nobles. In the major period of exploration in Renaissance Europe, absolute monarchies did sponsor much of the exploration and colonization of America. There are several reasons. First, exploration is an expensive long term investment, and thus a rich monarchy is well-placed to finance it. Second, much of the purpose of exploration (with a sideline in piracy!) was motivated by commercial and military rivalries between countries, the precondition for which, is to a degree a strong sense of nationhood and central coordinating power.
Indeed, I believe thanatassa's evaluation above is quite valid-- exploration/geaographical expansion is not a phenomenon linked or correlated to absolute monarchy; apart from the examples quoted above, id also like to add that the largest and most powerful modern colonial empire was the British empire which really expanded circa mid 19th century, after Britain itself underwent major political reforms and became increasingly a constitutional monarchy.
I also think that your question relates (as I see it) to the linkage between the 'Reconquista' of Spain and Christopher Columbus's voyage of exploration, commissioned thereafter in the same year 1592--am I right? So, in that case there might be some confusion at work.