Yet pity for a horse o'er-driven, And love in which my hound has part, Can hang no weight upon my heart In its assumptions up to heaven;
And I am so much more than these, As thou, perchance, art more than I, And yet I spare them sympathy, And I would set their pains at ease.
So mayst thou watch me where I weep, As, unto vaster motions bound, The circuits of thine orbit round A higher height, a deeper deep.
Tennyson loves his overworked horse and his dog, and this love doesn't interfere with his higher goals.
He's setting up an analogy: if he can love his animals without it interfering with his journey to higher things, then Arthur's spirit loving him shouldn't stop him (Arthur) from getting to higher things in Heaven.
It's kind of like one of those analogies you see on the SAT—Tennyson : a horse or a dog :: Arthur : some kind of heavenly creature. And Tennyson's better than a horse or a dog (we hope).