They didn't...Catullus 37 and 39 accuse the Iberians of rinsing their mouths with urine, as does Diodorus 5.33, who narrows it down specifically to Celtiberians, whom he accuses of both bathing and washing their mouths with urine. Strabo says the same thing about the Celtiberians at 3.4.16. But the common theme in these authors is that this is not only a foreign practice but a decidedly barbaric one--Catullus' accusations make for some of his funniest invective, and both Diodorus' and Strabo's testimony is nestled in with various slanders against the Celtiberians. Romans did not do this, they accused barbarians, and particularly a specific nation of barbarians, of doing it.

In point of fact the Romans were quite keen on dental hygiene. Pliny mentions toothpastes, usually made of ground pumice or something similar, with which Romans polished their teeth in the morning, and Horace mentions breath-freshening lozenges--Pythicus actually had an entire exposition on dental powders. The Romans did make use of urine for cleaning purposes, but not in bodily hygiene, which they found repulsive. Urine could be collected in chamber pots and tanks to be used in cleaning clothes by, for example, the massive public laundromats at Pompeii. The idea is that the urine, when stored, eventually breaks down into ammonia, whose cleaning powers were well-recognized by the Romans--Vespasian famously attempted to place a vectigal on urine collection, to the amusement of later authors