The short answer is they didn’t.

I am by no means a Romanist, but this question touches obliquely on two areas of history that are close to my heart - urban history and the satires of the British restoration.

Ben Wilson’s excellent Metropolis explores the history of the global city through a series of sections, each devoted to a particular ancient and modern city and some form of urban infrastructure or institution exemplified in both. The second of these focuses on Rome and its baths, and the crucial social role they played.

One key insight is that baths were hugely versatile spaces. They were places to keep fit, to socialize, to relax, to take treatment for various illnesses and injuries, and also served as a sort of informal public forum. One thing they weren’t was a place for getting clean. The water might be hot (or tepid, or cold, or all three at the finest baths), but it wasn’t very clean. It also generally wasn’t changed regularly, and after a while a cloudy scum of skin oil, dirt, and assorted crud would float to the surface. Several Roman writers reflected on the irony of seeking health at a place where people go to, more or less, marinate in other people’s skin goo.

If you were very wealthy, you could wnjoy a somewhat more sanitary dip by having a private bath installed in your home. A private bath was the sine qua non of domestic sophistication, and sure to bring your friends running with the promise of a relaxing soak sans direct contact with Severus the fishmonger’s armpits. With private baths such a potent status symbol, it’s no wonder that a talented Roman businessman named Sergius Orata became one of the wealthiest men in the empire mostly by installing them in luxury villas. Fun fact: he is also generally considered the inventor of oyster farming.

Skipping ahead a short fifteen hundred years, a fair number of Roman baths were still in operation around Europe at the middle of the eighteenth century, and not much had changed. The hilarious Expedition of Humphrey Clinker is a 1777 epistolary novel recounting the travels of a grouchy Scott around the British isles, seeking treatments for his various pains and afflictions. He visits the baths at Bath, and makes much the same disgusted comments as his ancient forbears. The book contains a solid chapter of gross-out humor, describing open pustules weeping into the bathing water while other patients drank it. Read this book after dinner.

So, in summary, your mental image of a typical Roman bath should probably feature fewer frolicking nymphs and more caked-on skin oil and assorted nameless schmutz. Cleaner options were available, but you had to either be very wealthy or score an invitation to rich friend’s home. If you’re an average Roman, your best bet to actually get clean is probably the pond.

Here's a post by /u/Celebreth that discusses the sanitary (or not so sanitary) conditions of Roman baths.