There are two things I think the choice of first-person adds to the poem: novelty, and a sense of movement.
The novelty factor comes from the point of view being, well, novel. Bodies of water can't talk, and they aren't usual narrators, so a reader will likely find a poem written from a brook's perspective novel and interesting. For example:
I chatter, chatter, as I flow
To join the brimming river,
For men may come and men may go,
But I go on for ever.
Here the brook does some unusual, interesting things. It "chatters", which is a human activity, but then immediately reminds the reader of its non-humanness by noting that it "flow[s]". The brook says it will "join" the river, which is not something that would happen with a human narrator. Personally, I laughed here, considering how yes, the brook's winding path is just to join a larger river.
The next two lines are more explorations of the novelty which can be taken from this point of view. The brook contrasts the normal lives of humans - they "may come" and "may go" - with itself, because it will "go on for ever". This is another compelling novelty; the brook, in its novel perspective as a natural feature, will go on forever.
A natural question is why a similar effect couldn't be achieved if the brook was simply described, without the first-person perspective. I would answer that by saying that the use of an "I" pronoun for these novel activities - remember, the brook says "I go on forever" - adds to the novelty. The "I" claims humanity, and the non-human contrast of what the brook does adds humor and novelty.
I could go on like this about every stanza of the poem, waxing eloquent about how the first-person perspective adds novelty. However I shall spare you that, dear reader, as I'm running out of ways to say "novel" and "interesting".
What do I mean by "a sense of movement"? The brook's narration contains many "movement" words. It "makes a sudden sally", "bicker[s] down a valley", "hurr[ies] down", "slip[s] between the ridges", "flow[s]", etc. These phrases create a relentless sense of movement, barreling with the reader through the poem and down the lines.
Making it an "I" who does these movements is a way of drawing the reader along with the movement. You are more likely to be dragged along by a bubbly, energetic friend than by a description of a bubbly, energetic friend. By declaring an identity with "I", the brook's movement words are more effective in pulling the reader along for the ride.
