I am not quite certain about how the Roman army gathered for a major campaign or battle during the Republican period, but I can give you an answer for the Roman Empire. Under the early Empire almost the entire army was garrisoned along the frontiers, called fines. Only a small force was permanently located deep inside the Empire: the Praetorians Guard, the Urban Cohorts, and the vigiles. All up, this was less than ten-thousand soldiers. The army was distributed like this because almost all warfare in this period was conducted outside the Roman Empire. There are inherent problems with projecting modern strategies back in time to the Romans, but the Roman army at this time did operate on a basis similar to the modern strategy of ‘forward defense’. The Roman command would gain knowledge of growing tensions or threats while they were still well outside Roman territory, and then would prepare a campaigning force by requisitioning legions and units called vexillationes from other frontiers. A vexillatio was a detachment of one-thousand men drawn from a legion. The vexillatio system ensured a commander could form a large campaigning force without either draining Rome’s other frontiers nor overly draining Rome’s economy. For example, we know that during the Dacian Wars legions were brought in from Germania Inferior and Raetia, and a vexillatio was drawn from a legion stationed in Judea.

This system only really worked to fight one or two major campaigns at a time, and you asked how the Roman army managed to fight in different places at the same time (although this did not happen often for the first two-hundred years of imperial rule). In truth, the Roman army did not manage to fight major wars in different places at the same time! The period known as the Third Century Crisis is evidence of this. In 235 C.E. emperor Severus Alexander was assassinated by his own soldiers near the Rhine River. What followed was the worst fifty years the Empire had so far suffered. Twenty-two emperors ruled for the next fifty years, with an average reign of fewer than three years. Constant civil wars saw imperial claimants take their armies off the frontier to fight against one another, leaving the frontier weak and exposed to invasion by Rome’s enemies. The weaknesses of this Roman military system are shown in instances like in early 268 when emperor Gallienus was occupied with the revolt of Aureolus near Milan. The Heruli and Goths were able to launch devastating invasions of Asia Minor and Greece at this time. Gallienus’ successor, Claudius II, would focus his attention on defeating the Goths in Greece, but this allowed the Alemanni to invade northern Italy and Queen Zenobia of Palmyra to seize Egypt. We can thus see how the Roman vexillatio system was incapable of fighting major wars in different places at the same time.

To rectify this weakness reforms had to be made to the military system. Emperor Diocletian made sweeping political and economic changes, but here I will focus just on some of his military reforms. Diocletian bolstered the Roman frontier defences by creating systems of defense that relied upon both ‘hard’ fortified positions, like large walled cities or fortifications, and quick moving reactionary forces that could intercept invaders. This system was supported by supply depots, factories, and garrisons spread throughout frontier provinces. Constantine reformed the Roman military further, by creating three types of armies: the limitanei, comitatenses, and the palatini. In the primary material these names actually varied over time, but the three used here are the most common used by historians for the sake of simplicity. The limitanei comprised roughly sixty-percent of the army, and were distributed along the frontier to guard against small-scale incursions and fulfil other border responsibilities. The comitatenses were mobile field armies garrisoned deeper inside the Empire. There were three comitatenses in the east and five in the west. The palatini armies were similar mobile field armies, but were garrisoned at the imperial capitals and accompanied the emperors.

In this new system, to gather a large army for a campaign or battle, the standard practice was for an emperor to take his palatini army towards the location of the campaign or threat, gathering one or multiple comitatensis armies along the way, and perhaps even drawing more soldiers from the regional limitanei. This practice is similar to the ‘forward defense’ I described earlier, however this new system was designed to be able to better target threats that had already penetrated the Roman border, as well as to protect the current emperors from usurpation, as they always had an army nearby. The drawback of this system was the borders were always significantly weaker. The limitanei were most likely able to deal with any threats of one-thousand or less enemy soldiers, but an invading force of significant size would not find it difficult to cross the Roman frontier. Was this system inherently better or worse than earlier? It is difficult to say. This same system was still in use during the fifth century when the Empire collapsed, but personally I see the explanation for Roman weakness in the fifth century as a result of the child-emperors which ruled, not from the military system.

As for the second part of your question, you ask how the size of the Roman army was possible. To be honest, I am not sure what part is actually confusing to you. The Roman army of the Empire numbered hundreds of thousands of men. Agathias claims that the army reached the extraordinary size of 645,000 men, while Zosimus claims a lower figure of 581,000. John Lydus gives a more reasonable figure of 390,000 during the time of Diocletian – an amount similar to the army under Hadrian. The Notitia Dignitatum indicates a size approximately around 450,000 before 378. These counts, whichever is most accurate, are large numbers of soldiers, but remember that they were spread over many thousands of kilometres of frontiers and many garrisons. The population of the Roman Empire was in the many millions, and while the data is very difficult to be certain on, some studies suggest that Roman military expenditure as a percentage of its total GDP was relatively similar to modern states.

Sources:

Yann le Bohec (ed.), The Encyclopedia of the Roman Army

Yann le Bohec and Catherine Wolff (eds.), Les légions de Rome sous le haut-empire: actes du congrès de Lyon (17-19 septembre 1998).

Williams, Stephen. 2000. Diocletian and the Roman Recovery.

Southern, Pat. 2015. The Roman Empire from Severus to Constantine.