Do you have any particular manpower crises in mind that you'd like explained? The late Roman/Byzantine era is a thousand years long, there's no single generalization that can explain every crisis in a satisfactory way. I'd be glad to help answer questions for the middle Byzantine period.

The essential difference is the the army of the Middle Republic was a militia army, where every male citizen (of a certain minimum property qualification) was obliged to serve, motivated by patriotism and the defence of his own home.

In imperial times, the army turned professional. This not only meant that soldiers would get much better pay and a retirement bonus, but also that joining the army now meant a 25 year commitment, moving to the frontier, not being allowed to marry, and being subject to the strictest discipline at all times.

This new army only drew recruits from the lower social strata (except, of course, for the officer corps, including centurions) and even then more from the frontier provinces, where a warrior-culture persisted long after decades and centuries of relative peace reduced this in Italy itself.

Moreover, the lower social class and provincial origin of the members of the professional army resulted in a much lower status of the army among civilians. Rather than something proud and patriotic, it came to be seen as something alien and potentially dangerous. This would only be reinforced by civil wars, when Roman armies could sack Roman towns, or by the breaking down of the organised tax and supply system in the 3rd century, when (unpaid) requisitioning became the only method for the army to supply itself.

Long story short: most civilians no longer wanted to join the army.

Add in problems with inflation reducing the true value of soldiers' pay, lower number of offensive wars reducing the chances for loot and self-enrichment, and possibly falling population numbers due to the plagues that started to periodically hit the Empire from the late 2nd century onwards and possibly climate change, and it's not starting to look good. Finally, towards the very end of the western empire, reduced tax revenue for the central government started to make it really difficult to recruit armies. This happened both because of barbarian peoples occupying valuable provinces (Africa, Spain, parts of Gaul) and because the rich land-owners in the west were apparently very good at getting out of paying their taxes.

In the late army (and occasionally in the early imperial army too) the authorities tried to ameliorate the situation through conscription. Communities often had to provide set numbers of men. Unfortunately, this too involved a lot of corruption, such as sending only the most useless and weak men they could find to meet the quota, or just paying the governor off to get out of having to send men.

All this helps explain why the so-called "germanisation" of the legions became a thing from the 4th century onwards. Outside the empire, willing recruits could still be found. Prisoners of war were also often conscripted into the legions, and defeated tribes had to provide recruits as a form of tribute. This used to be held up as an explanation for why the legions became weaker and to explain Rome's fall, but nowadays this isn't really seen as a bad thing and most historians argue that the German recruits were good soldiers and helped preserve the Roman empire. After all, the Romans had always drawn such "barbarians" into their army as auxiliaries, even in the days of Augustus.

The Byzantine period is an entirely different story... or possibly not. I don't know enough about it to even start on an answer, sorry.

All that said, this is and remains a question about which there is much debate. There's even debate about the exact nature of this manpower shortage. Was it actually impossible to find recruits, or are the complaints we find here and there in the sources blown out of proportion? At a certain point there wasn't much of a Roman army anymore in the west, but exactly which point that is remains open to discussion.

Info mostly drawn from Goldsworthy, Fall of the West and Complete Roman Army

It didn't. The Late Roman Army was massive, even compared to the army of the Principate. The size of the late army is consistently estimated to be somewhere between 400,000 and 600,000 strong, by historians such as Agathias, Zosimus, and many modern historians too. On a similar vein, the late army, even into the final years of the Western Empire, was the victor of the vast majority of its battles. Justinian experienced trouble with funding the army, rather than manning it.