The way I like to explain it is that there aren’t two reference frames in the classic version of the Twin Paradox; there are three. It’s easiest to describe if we imagine the traveler as a hitchhiker, rather than in command of his own vehicle. That is, suppose that Larry stays home on Earth, while his twin Jerry hitches a ride with Xyzzy the alien, who happens to be passing by at 0.9 c. Later, as they’re passing alpha Centauri, Jerry gets homesick, and hitches another ride with Quux, who’s coming back the other way at 0.9c.
We now have three reference frames: Larry’s, Xyzzy’s, and Quux’s. Jerry doesn’t have his own reference frame, because he changes from one alien’s frame to the other’s. All three reference frames will agree that, when the twins meet up again, Jerry is younger than Larry, and by how much. They disagree about points in between, but that doesn’t actually matter.
From Larry’s point of view, Larry was always aging at the normal rate, and Jerry was always moving fast relative to him, and so Jerry was always aging slower.
From Xyzzy’s point of view, Xyzzy was always aging at the normal rate, and Larry was always moving fast relative to him, and so Larry was always aging slower than Xyzzy was. Meanwhile, for the first part of the trip, Jerry was at rest relative to Xyzzy, and so for that part of the trip, he was aging normally, too. But then when Jerry got homesick, he had to travel really, really fast to catch up to the already-fast-moving Larry. And during that time, he ages even slower than Larry does, so much so that by the time he does catch up, Larry is older.
Likewise, from Quux’s point of view, except that for him, Jerry’s time of aging really, really slowly is the first part of the trip, when he’s with Xyzzy.