According to Lehmann's 1951 Proto-Indo-European Phonology (which is admittedly somewhat outdated but is the best resource I have on this), there are three main reasons to postulate some sort of reflex of laryngeals in Vedic Sanskrit.
One, as Mellifluous mentions, is aspiration when laryngeals came after a stop, as in tíṣṭhati "stand" from the zero-grade of *steh₂. But this could have happened well before Vedic.
Another is lengthening in certain compounds. For example, abhí "to, for, on" generally has a short ĭ in compounds. But certain roots cause the vowel to lengthen: abhīsat "surpass" < *h₁es-. These roots also often have a lengthened augment.
Finally, and most convincingly, Vedic meter sometimes requires a long vowel to be read as two short vowels in hiatus; this is what the quote in your question is talking about. For example, yānti "go" (from *yeh₂) often needs to be read as three syllables, implying it was originally *yaHánti.
In particular, superlatives in -iṣṭha- tend to show this hiatus when applied to stems ending in a long vowel: déṣṭha "most generous" < dā < *deh₃ has to be read as *daHiṣṭha. And one general tenet of the laryngeal theory is that all roots in PIE consisted of two groups of consonants; stems which end in a long vowel, like dā, generally come from that second group consisting of a single laryngeal.