That's Edge of Forever, there, Redshirt 😉
The obvious answer for why people go back in time to just before shit gets real is for the drama, but if I were to try to handwave a physics reason for it, I'd say that the energy surrounding a major historical event (Tulsa, Pompeii, Kennedy assassination, etc.) draws the time eddies towards it. Obviously doesn't apply to quiet, personal situations but whatevs.
The speech during the burning, and the song during Leti's walk, were the same passage. One spoken and one sung. But I don't know the piece and I don't know if the poetry preceded the song lyric or vice versa.
Oh I guess it was a poem first.
And inspired the show's soundtrack composer to write the song at the end.