As commenters have noted, Arwen was not originally intended to encounter Frodo in this scene. Peter Jackson substituted her for Glorfindel, probably for a few reasons (the romantic subplot makes sense, as does removing an appearance of an otherwise redundant character1).
One possibility is that Arwen is using an elvish spell, like the one she used to rouse the river and foil the Black Riders. Considering that magic in Lord of the Rings is a very specialized thing, I find this unlikely, but it's most consistent with the dialogue. The "grace" she refers to may be this magical gift, or it may refer to her elven immortality, which she may have shared with Frodo to help him survive long enough for Elrond to heal him fully.
The more logical explanation is that Arwen, as Elrond's daughter, was simply a better healer than Aragorn and not as good as her father. The "grace" would then be her healing skill, which she used to stabilize Frodo long enough to get him to Elrond, and the white light we see is a visual representation of Frodo going into a coma as a result of his poisoning. This makes more sense in the context of the world, but is inconsistent with the dialogue and actions of Arwen.
Desolation of Smaug confirms this part of the theory. When Kíli is suffering from the Morgul poison, he briefly sees Tauriel standing with a bright white light behind her. Since Tauriel is earlier confirmed to be a Silvan elf, and therefore had no ancestors in Valinor, there's no other explanation for this shared phenomena (unless you assume that Frodo was in love with Arwen which, outside of fanfiction, isn't going to hold water)
1 Funny, related story: While he was writing the book, Tolkein was part of a writer's club called The Inklings, which included C.S. Lewis (Of The Chronicles of Narnia fame). All members of the group were writers, and they would often read their unfinished drafts to one another for critique. During one such reading of an early draft of one of the books, Inkling Hugo Dyson (A vocal critic of Tolkein's story) loudly exclaimed "Oh no, not another fucking elf." The exact quote is disputed, but the sentiment remains: too many friggin elves.