Sylvia Plath is known as a confessional poet, which means that she wrote highly personal, detailed, and emotional poems. For this reason, we might want to assume that the speaker of "Ariel" is Plath herself. (Plus, we also know from Plath's husband that she really did have a horse named Ariel who once took her on a wild ride. Check out "What's Up with the Title?" for more info.)
But, all this being said, we've got to admit to ourselves that we can't ever really know how accurately Plath represented herself in her poetry. Did she actually become one with her horse Ariel? What would that even look like? For these reasons, we don't refer to the speaker as Plath herself.
Instead, here's what we do know about our speaker: she (and we're just assuming it's a she) goes through a transformation. When the poem and the horseback ride begin, our speaker's holding on for dear life. But, by the end of the poem, she's loosened—even let go of—her grip. The language at the end is steeped in deathly images. The speaker is "suicidal"—but not in an ordinary way. The speaker metaphorically kills her old self so that she can be born again as a powerful horse-woman. (So cool, right?) The speaker has transcended ordinary human life and summoned the power of nature. She becomes "at one" (26) with her horse's ride into the rising sun. She lets go of her inhibitions and has an Experience (with a capital E).
Not bad for an early morning horseback ride.