Of course, in order to understand this, you have to know what "accidents" and "substance" are. Whatever the senses perceive of a thing are the accidents of that thing. They are not the thing itself but merely the perceptible qualities or characteristics of the thing. The substance, however, is the thing itself, or rather, the essence of the thing.
So, take for example the bread used in Mass. The accidents of it are: roundness, whiteness, crispiness, bread-like smell, bread-like taste. The substance of it is: "bread." Our senses perceive the accidents; only the mind knows the substance.
In every case in the universe but one, when the substance changes, the accidents of it change too since the accidents are attached to (or, exist in) the substance. For example, when a caterpillar becomes a butterfly, it also changes in outward appearance b/c, well, butterflies have different outward characteristics than caterpillars do.
Only in the transubstantiation of the Eucharistic elements does the accidents remain even though the substance changes. Maybe an illustration will be helpful. I saw a magic trick once where a man in a black costume stood in the middle of the stage. Some assistants pulled up a curtain around him. There was smoke and flashes of light. When they dropped the curtain to the floor there stood a woman in the same black costume. Transubstantiation is kinda like that. The "costume" of bread is suspended ("in mid air" so to speak) while the underlying substance (or thing that wears the costume) is changed.
I hope that helps you to make sense of this mystery. It is universally accepted that all created things have accidents and substance. The only quarrel is over whether or not God desires to suspend the laws of the universe in this one instance in order to be substantially present in the Eucharist.
There are two articles that I highly suggest if you would like to learn more about transubstantiation:
- F.J. Sheed, Theology for Beginners, Chapter 18: "Transubstantiation"
- Theology for the Laity: "Transubstantiation" by Fr. Paul A. Duffner, O.P.