In other words, when an English speaking person would say "I see" meaning "I understand what you're saying", is it natural in classical Latin to say Intellego, as in, maybe even more than once? If not, is there such an idiom at all?
Intellego. and Teneo. are both used as stand-alone replies in Plautus and Terence with the sense of "I understand" or "I get it":
Both of the following are examples from the Andria of Terence. The second one in particular is useful because you can clearly see that it is being used as an interjection by the second speaker to reassure the first-- she literally interrupts him to say it.
"SI: rogitabam 'heus puer, dic sodes, quis heri Chrysidem habuit?' nam Andriae illi id erat nomen.
"PA: propera. atque audin? verbum unum cavĕ de nuptiis, ne ad morbum hoc etiam . .
Examples from Plautus:
"LYS: Primum ego te porrectiore fronte volo mecum loqui;
stultitia est ei te esse tristem, cuius potestas plus potest.
probum te et frugi hominem iam pridem esse arbitror.
"EP: Hoc quod actum est. egomet postquam id illas audivi loqui,
coepi rursum vorsum ad illas pauxillatim accedere,
quasi retruderet hominum me vis invitum.
NB: I cannot find any examples in the PHI of capio being used as a stand-alone phrase/interjection in classical Latin with the meaning "I understand."