The poem begins with a metaphor: the speaker says she isn't asking to be young again, or to "delay / [...] time's irreversible river." In other words, the speaker understands time as a river that flows in only one direction: toward old age and, eventually, death.
By comparing time to a river that will swallow up the "jewelled arc of the waterfall" (i.e. her life—more on this in a second), the speaker illustrates the sheer force of time, which stops for nothing—and also acknowledges that the process of living, aging, and dying is natural. She wants to appreciate the beauty and brevity of her one life, while also not fighting the flow of the way things are meant to be. In other words, she is trying to grow old gracefully and confront her mortality with acceptance rather than fear and bitterness.
The waterfall itself is a metaphor for the speaker's own life: both breathtaking and fleeting. The imagery of the waterfall's "jewelled arc" immediately suggests that there is a trajectory to the speaker's life: a beginning, middle, and end to the speaker's life. But this transient "arc" is also "jewelled": something to be admired and wondered at, not taken for granted or resisted.
The poem's structure mirrors this bright flow. For instance, the speaker's enjambment across lines 1-3 (and throughout much of the poem) imitates the ceaseless movement of time's metaphorical waters, which cannot be "delay[ed]" and which eventually claim each and every life.
And the musicality of the consonance and assonance in these lines ("in the rising of time's irreversible river / that takes the jewelled arc of the waterfall") helps to emphasize the speaker's understanding of life's fleeting beauty.