What happened that cause the size of armies to go from hundreds of thousands in the time of Alexander the Great to just mere thousands in the time of the battle of Agincourt?

They weren't likely in the hundreds of thousands. Hyperbole in military statistics should be expected, not surprising. When you hear of a big number, let's say Herodotus' claim that Xerxes' forces numbered some 2.3 million men, you should ask "where does that number come from? Does it make sense?"

Numbers in ancient and medieval texts are notoriously problematic-sometimes entirely plausible and seemingly trustworthy, according to our modern ideas of what is possible or not under certain sorts of conditions; sometimes entirely incredible, yet at the same time not always to be cast aside or ignored, perhaps having a symbolic or indicative value even if not providing an exact arithmetic count. Fulcher of Chartres, an eyewitness, reports that the forces of the First Crusade at Nicaea in 1097 numbered 100,000. He reports further that 5 million noncombatants were also present! By contrast, another eyewitness, Raymond of Aguilers, reports that by the time the army reached Jerusalem it had been reduced to just 12,000. [Haldon 2011]

Alexander's army has a good chronicler in Arrian, but there are still problems. If you look at, say, accounts of the Battle of Granicus, we have plausible numbers for the Macedonians, perhaps 20-30,000-- but the numbers for the Persians vary wildly and are generally suspect, for example Arrian tells us that the Macedonians killed 18,000 of the 20,000 Persian mercenaries. Scholars like Peter Green read this and are unconvinced. Similarly, Marcus Junianus Justinus tells us that there were 600,000 Persians; its safe to assume that there weren't. How few were there? Reasonable estimates are more like 30,000.

Some estimates (Delbruck) go much lower-- and I wouldn't argue against 10,000 on either side. One aspect of Arrian and other accounts that seems credible is the personal involvement of Alexander and his companions in battle, as a meaningful military unit - rather than just as a commander. If we accept the importance of these individuals and small units as accurate, then we have to be thinking of relatively small numbers.

Its notable that when later in antiquity we have chroniclers who were clearly in a position to know -- like for example Procopius, who was a secretary to Belisarius-- the numbers often get smaller. Our reports of army size often get smaller as we get closer to the present and have informants who are closer to the data. As we go back in antiquity, we're often looking at just one source with a bias towards the heroic rather than the logistic, the tendency to inflate army size can be significant.

Earlier in antiquity, with chroniclers who have reason to exaggerate or who have second hand information we find the numbers that get very large. Modern scholars have done a lot of work to try to gauge what sort of numbers could actually make sense. Most estimate Alexander and Darius' armies in the tens of thousands, rather than the hundreds of thousands. Scholars are beginning to explore quantitative techniques for estimating army size, looking for clues, but its early days. There's a lot of aggressive interpretation of scraps of data, for example the amount of transport that would be required to adequately supply an army of given size; while these techniques have their own challenges -- a lot of interpretation can hang of very little data. In general the supply and transport of massive armies has a commensurately large logistics footprint, and it's quite often that we see a claimed size that just doesn't match what is the likely available food, population, or transport capacities.

Then you have the problem: "if you know that the claimed number must be wrong-- how do you get to a more plausible number". Arriving at statistics for wars and violence isn't dispassionate census taking; calculating violence is still a challenge: there was quite recently a controversy between demographers and epidemiologists trying to assess something that's in plain view-- civilian deaths in Iraq; that these numbers differed in some cases by hundreds of thousands gives a measure of just how difficult this subject can be.

See:

Conor Whately. “Some Observations on Procopius' Use of Numbers in Descriptions of Combat in Wars Books 1–7.” Phoenix, vol. 69, no. 3/4, 2015, pp. 394–411.

Charney, Michael W. “A Reassessment of Hyperbolic Military Statistics in Some Early Modern Burmese Texts.” Journal of the Economic and Social History of the Orient, vol. 46, no. 2, 2003, pp. 193–214.

HALDON, JOHN, et al. “Marching across Anatolia: Medieval Logistics and Modeling the Mantzikert Campaign.” Dumbarton Oaks Papers, 65/66, 2011, pp. 209–235.

McCoy, W. J. “Memnon of Rhodes at the Granicus.” The American Journal of Philology, vol. 110, no. 3, 1989, pp. 413–433.

Bosworth, A.B. "Conquest and Empire: the Reign of Alexander the Great", (Cambridge University Press:1988)

Dykstra, Robert R. “Quantifying the Wild West: The Problematic Statistics of Frontier Violence.” Western Historical Quarterly, vol. 40, no. 3, 2009, pp. 321–347.

Tapp, C., Burkle, F.M., Wilson, K. et al. Iraq War mortality estimates: A systematic review. Confl Health 2, 1 (2008). https://doi.org/10.1186/1752-1505-2-1

F.P. Lock, "The Rhetoric of Numbers in Gibbon's History", (University of Delaware Press:2012)