Poets can share with you a mood or perspective of theirs in fewer, more pungent words. Both their mood / perspective and the unexpected way they use words can (in a mild but pleasing or even exciting way) change your life, or at least your day.
You might say poetry is part way between music and prose -- it leverages its open-ended-ness to be more evocative than a literal description.
Or better yet, it's like the difference between paintings (or between photos) where you don't see a literal representation of the subject, instead you see the subject in a new way.
My bachelors was in general literature with an emphasis in modern English poetry. A lot of poetry bores me. Part of the problem is it's too dated, or the poet is too obscure or too literal.
You said not to recommend poems, but here are my two favorites: "The Love Song of J. Alfred Prufrock" T.S. Eliot (1915) and "To His Coy Mistress" Andrew Marvell (1681).
Also "Desolation Row" Bob Dylan (1965). In "Visions of Johanna" (1966) Dylan describes looking at a woman facing him in a dimly lit room: "the ghost of electricity howls in the bones of her face." It can tickle someone who loves words to unpack that.