This passage of Scripture relates to Roman Catholic transubstantiation because it affirms that Jesus Christ is our source of spiritual nourishment. He is our spiritual food. He, the substance, is the same substance that is made present by faith at the meal of the New Covenant.
Transubstantiation does not involve the eating and drinking of Christ in a "spiritual" manner. That language is distinct from the language of Catholic dogma. Transubstantiation takes place when the priest pronounces the bread and wine to be Jesus Christ's actual body and blood (the change is not in the accidents, but substance).
What matters in Roman Catholic theology is the participation in that physical substance. However, this is contrary to the way that the Scriptures speak concerning the participation of those ancient people in the meal and water during the Exodus. Moreover, Paul stated that Jesus Christ was the Rock (1 Corinthians 10:3-4), yet He never underwent a process of literal petrification.
The Israelites "ate" and "drank" Christ, which must be spiritual and not physical. No transubstantiation took place during the Old Testament. We partake of Christ in the same spiritual way that the Jewish people did, which is by faith.