Just to clarify michelle's "put them on" answer (which is correct), "to put an article of clothing on" means to dress in that article. A single English verb for this would be to "don" an article of clothing, but that expression is slightly formal/antiquated these days.
I'd disagree with TRomano's comment on the implied quickness, however. The "whirl of her skirts" bit could potentially imply quickness, but simply saying "on went her jacket" does not. In fact, I'd question whether the "whirl of her skirts" is intended to imply quickness -- it could imply a joyful, lighthearted motion. This also works with "fluttering" down the steps, and the sparkle in her eyes.
Further, I absolutely disagree with the assertion that one wouldn't use "off went her [x]" in a seductive way... A long string of those statements tied together would leave me feeling with a slow, steady, almost rhythmic progression. Again, though, that's due to the use of the phrase in a specific way; I'd say that by itself, "on went [x]" doesn't imply slow or fast action.