Upon marriage with Aragorn, Arwen became mortal, and began to age and die the way that Númenöreans do. She then chose "resign her life" 30 years early.
The scene that you are quoting is showing Arwen resigning her life. At this point Arwen is a Númenórean, and "good Númenóreans" generally will willingingly do this rather than "clinging to life" and waiting for themselves to die of old age.
The first approach of "world-weariness" was indeed for them a sign that their period of vigour was nearing its end. When it came to an end, if they persisted in living, then decay would proceed, as growth had done, no more slowly than among other Men. Thus a Númenórean would pass quickly, in ten years maybe, from health and vigour of mind to decrepitude and senility. In the earlier generations they did not "cling to life," but resigned it voluntarily. "Clinging to life," and so in the end dying perforce and involuntarily, was one of the changes brought about by the Shadow and the rebellion of the Númenóreans; it was also accompanied by a shrinking of their natural life-span.
Unfinished Tales - Christopher's commentary to "The Line of Elros"
Aragorn resigned his life 24 years early, and Arwen resigned her life the year afterwards, which was 30 years early for her.
We have these calculations as performed by Tolkien, and confirming that Arwen did in fact die, but first a bit of context to understand his math.
Elves grew one year every three years for 72 years, or until they turned 24. They then grew one additional year every 144 years for the next 13,824 years, or until they turned 120. After that they began to fade. But elves also matured faster than Men, by a rate of 4:3, so an Elf who was 24 (i.e. 72 years) would be the equivalent of a mortal 18-year-old
Númenöreans grew one year every year for 24 years, followed by one additional year every three years for the next 186 years, when they would be 86, and then they would decline for another 24 years until death, living a total of 234 sun years.
When Arwen marries Aragorn she was thus 43 "Elvish years" old (24 * 3 + 19 * 144). This was then converted to the equivalent of a Númenörean who was 43 "Númenörean years" old (24 * 1 + 19 * 3), who would have lived off 81 of their allotted 234 years and thus have another 153 sun years left.
So when Arwen choose to resign her life 123 years later (having technically lived 2,800 sun years, but now the equivalent of a Númenórean who had lived 204 years, or a regular human who had lived 84 years), she would have had another 30 years left.
The case of Arwen. Taking her birth as TA 241, she will then be “full-grown” in TA 313 (241 + 72). In 2951, when she first meets Aragorn, she will be (in Elven Growth- and Life-years) 24 + 18⅓ (nearly); (2,951–313)/144 = 42⅓ = mortal equivalent 31¾. Aragorn was only 20.
In 3019, when they were married, she would have aged very little and would be nearly Elvish 43 (24 + 3,019–313) = mortal equivalent 32–3. But Aragorn would have lived 88 years and 4 months. His “age” would however be about “45”. (See the note below on the Númenórean life-scale.)
At marriage Arwen became “mortal”: she would then join her husband’s scale of “expectation of life”. ☞ This would not alter her “age” of 43 = approximate mortal equivalent 32–3. But for the purpose of reckoning her expectation of life (as a mortal), she would count as having lived 81 years (24 + (19 × 3)), and her further “permitted life” would be about 153 years (to total 234). She might have lived on to about Fourth Age 151. Aragorn as 88 at wedding would have a permitted life of 146 more years and could have lived to about 4A 144. When Aragorn “resigned life” in 4A 120 he thus resigned 24 years of life. He had lived 210 years and was already within his “decline”. Arwen was reckoned as 203 years at that time and also in the beginning of her decline. Aragorn resigned on the day of his birth, March 1, 4A 120. Arwen apparently “resigned” life and died on Cerin Amroth on March 1 in the following year, at Númenórean age 204 (mortal equivalent = 84). So she was now and felt. (Had she remained Elvish she would have been only 3,020 – 241 + 1 + 120 = 2,800 Sun-years old, giving an Elvish age of 24 + (2,800–72)/144 = almost exactly 43 (mortal equivalent about 33).
The Nature of Middle-earth - "Elvish Ages & Númenórean"