What does it mean that Krishna's body is spiritual?
All the bodies we see in this world are temporary and made of matter. They're material bodies.We're different from our material bodies. Our bodies are structures composed of material elements—solids, liquids, gases—but our self, the spirit animating the material elements, is different. When the self, or spirit, leaves the body, the body ceases to function; it has no value on its own. The spirit is the real and valuable component of the body. We can't see it, but we know it's there by its symptom—consciousness.
As matter has its characteristics, spirit does also. The characteristics of spirit are sat, chit, ananda—eternity, knowledge, and blissfulness. Krishna is known as sat-chit-ananda-vigraha—His spiritual form is the embodiment of eternity, knowledge and bliss. Whenever He appears within the material worldHe does so in His spiritual body. This means Krishna's body isn't temporary or made of matter; it always exists, and it doesn't degrade over time.
Brahma-samhita, 5.1: "Krishna who is known as Govinda is the Supreme Godhead. He has an eternal blissful spiritual body. He is the origin of all. He has no other origin and He is the prime cause of all causes."
We also have our own original, spiritual forms. Right now, they're covered by material elements. We are not our bodies, but Krishna is His body. There's no difference between Himself and His form.
Bhagavad-gita, 4.5-6: "The Personality of Godhead said: Many, many births both you and I have passed. I can remember all of them, but you cannot, O subduer of the enemy! Although I am unborn and My transcendental body never deteriorates, and although I am the Lord of all living entities, I still appear in every millennium in My original transcendental form."
In the purport to that verse, Srila Prabhupada says,
"Although He appears in the same transcendental body and is Lord of the universe, it still appears that He takes His birth like an ordinary living entity. And although His body does not deteriorate like a material body, it still appears that Lord Krishna grows from childhood to boyhood and from boyhood to youth. But astonishingly enough He never ages beyond youth. At the time of the Battle of Kurukshetra, He had many grandchildren at home; or, in other words, He had sufficiently aged by material calculations. Still He looked just like a young man twenty or twenty-five years old. We never see a picture of Krishna in old age because He never grows old like us, although He is the oldest person in the whole creation—past, present, and future. Neither His body nor His intelligence ever deteriorates or changes. Therefore, it is clear that in spite of His being in the material world, He is the same unborn, eternal form of bliss and knowledge, changeless in His transcendental body and intelligence."