Everyone recognizes that there is more to what a thing is than its physical attributes. Otherwise a thing’s attributes would not be related to one another, and we could not distinguish particular things at all; the world would be a chaos of various attributes.
Suppose outside your window there is a tree with a blue jay in it. Your senses perceive a variety of physical attributes: brownness, greenness, and blueness, rough texture and soft appearance, chirping and rustling. All of these sensible perceptions reach your brain simultaneously, but you do not perceive it all as a jumble, nor do you group it all together and think of it as a single continuous entity "bird-on-a-tree." You understand that some of these properties (blueness, softness, chirping) are united to one another and constitute one entity, while the others (roughness, brownness, greenness, rustling) constitute another entity. Above and beyond the sensible attributes you perceive, you recognize something more: distinct entities uniting in themselves these attributes and not those.
Scriptural accounts tell us about unions of attributes which seem to be entirely miraculous—for example, angels appearing as human beings, sometimes so much so that they are taken for men. There would be nothing obviously miraculous about any specific attribute of such a vision. The hair, skin, teeth, eyes, fingers, and toes would all look and feel perfectly normal. But there is no human nature underlying the effect. When the miracle ceases, the man appears to vanish before our eyes. There is a union of attributes, but an entirely miraculous one, with no natural principle of unity—no substance—holding it together, as in the case of a real man.
Catholics believe the same is true of the Eucharistic elements after the consecration. The physical attributes of bread and wine remain, but in a manner similar to the physical attributes of human flesh in an angelic appearance. These appearances are held together not by a natural principle of unity, as in the case of ordinary bread or a real human being, but by a divine miracle. Were the miracle to cease, the appearances of bread and wine would (Catholic teaching holds) vanish before our eyes.