My favorite devotional book, Imitation of the Sacred Heart, tells us that we need to operate in such a manner that we presume everyone else is holier than ourselves to crush all pride.
For the saints, though, it wasn't a mere presumption. They really believed that. It wasn't just something "make believe" that they convinced themselves of to gain humility. They saw their own sinfulness, realizing that nothing good in them was their own, but was from God, and that all that they contributed to their relationship with God was the bad stuff. Combine that with deep charity towards their neighbor, and they really believed it.
If you were to tell a saint, "You're a saint," they would be disgusted and offended by the suggestion.
I recall a story told from the pulpit. I can't recall if it was St. Charles Borromeo or St. Philip Neri who was the main protagonist. There were rumors going around about how a certain religioius house had a true saint there. So he went to visit and asked, "May I speak to the saint please?" And the "saint" stepped forward and said, "I am he." And our Saint immediately turned around and left.