Who is the "I" in "The Brook" and what does he do throughout the poem?

The "I" in the poem is the eponymous brook. In other words, the poem is narrated by the imagined voice of the brook. A brook is a small stream.
Throughout the poem, the brook describes its journey through different landscapes and environments. In the second stanza, for example, the brook describes how it hurries down hills, slips between ridges, and travels beneath "half a hundred bridges." In the third stanza, the brook describes how it eventually joins with a "brimming river." In the subsequent stanzas, it continues to describe its journey through different landscapes and environments, such as "brambly wildernesses" and "many a fairy foreland set / With willow-weed and mallow."
Repeated throughout the poem is the refrain, "For men may come and men may go, / But I go on for ever." The implication here is that the brook, or rather the water which comprises the brook, will travel through thousands of tributaries, streams, and rivers in the course of an average human life. This perhaps reminds us of how fleeting human life really is and how indifferent nature is to human life.