The poem uses quite a lot of anaphora (and general parallelism), which lends rhythm and insistence to the speaker's argument.
The speaker starts more than half of the poem's lines with the word "I," in fact, calling readers' attention again and again to their presence. Take lines 3-4:
I am not there,
I do not sleep—
The poem's end then choices these lines:
I am not there,
I did not die.
All this repetition conveys the fact that the speaker hasn't disappeared from the world in death. They're just as present on the earth as they are in this poem.
The speaker repeats the longer phrase "I am the" in lines 5-8, 10, and 12. Again, this emphasizes the idea that the speaker still exists, that they still "are." All of this repetition also simply makes the poem feel very declarative and confident. Listen to the anaphora in lines 5-8, for example:
I am the thousand winds that blow
I am the diamond glints in snow
I am the sunlight on ripened grain,
I am the gentle, autumn rain.
The speaker is emphatic about their transformation. Thanks to all this anaphora, the poem feels powerfully self-assured.