They didn't.

The elite Roman education structure had some military education built into it: people like Cicero served as minor officers in campaigns, and there was the possibility for making a military career (rather than a civil career like Cicero's), but there was no real standard of military competency necessary for election to the offices that held imperium. To the best of my knowledge, Cicero had no military service between the Social War (90-88) BCE, where he served as a camp orderly, to his proconsular command in Cilicia in 51 BCE, for which he demanded (and was refused) a triumph. We also see people like Scipio Africanus winning elections to the consulship they are not on paper qualified for by age alone, and yet some of those people excelled in military endeavours despite their lack of experience.

Nevertheless, the lack of military experience could be a talking point for a candidate. Sallust, in his history of the Jugurthine War, put this in the mouth of Marius as part of a campaign speech:

Compare me now, fellow citizens, a 'new man,' with those haughty nobles. What they know from hearsay and reading, I have either seen with my own eyes or done with my own hands. What they have learned from books I have learned by service in the field; Think now for yourself whether words or deeds are worth more.

The "new man" thing is Marius pointing out he doesn't come from a noble line - his whole argument for election is his military experience, and he's contrasting that to others that are arguing they can win the war based on their lineage and their reading of manuals. The fact that Sallust put that speech in Marius' mouth indicates that the lack of military experience was indeed something Romans often overlooked when selecting their magistrates.

And in fact, Rosenstein has argued in "War, Failure, and Aristocratic Competition," Classical Philology, Vol. 85, No. 4 (Oct., 1990), pp. 255-265, that the Roman political system was set up in such a way that commanders would not usually be blamed for military failures. Instead they, and other Romans, blamed the incompetence or cowardice of the the soldiers. And he shows that such failures had few consequences for the losing Roman generals, as the majority continued their political careers.

Goldsworthy, in his pop history In the Name of Rome, concluded (and this was his only interesting conclusion in that book, as good as his academic work is) that the success or failure of a Roman general depended far more on how he prepared his troops with training etc., rather than execution of smart tactics. I don't know if that's true, or if Goldsworthy stands by that conclusion now, but the body of evidence seems to me to show that usually military competence wasn't a concern for the Roman electorate. It certainly could be in special cases, but those were special.

To add to u/LegalAction's answer where they mention manuals, there did exist textbooks on military matters. A short list of Roman examples (with dates):

Cato the Elder (2nd century BC)

Onosander (1st century AD)

Frontinus (1st century AD)

Arrian (2nd century AD)

Polyaenus (2nd century AD)

Vegetius (4th century AD?)

Apart from Cato, these are all from the imperial period, when the Roman army was more professional and the military command structure was a little more permanent, so perhaps there was a need for greater expertise. Nevertheless, there is good evidence that similar kinds of texts existed in the Republic, when the army was being led by elected officials for a year at a time. In one speech, Cicero complains that young men of his day are more likely to read about military exploits than experience them for themselves (Pro Fonteio 42-3), and in another he praises Pompey for doing just that - learning by experience rather that theoretical study (de imp. Gn. Pomp. 27-8).

The implication here is that only mediocre leaders would consult such works, rather than relying on their own natural virtues or talents.