The fox's secret, that "anything essential is invisible to the eyes," serves as a sort of "thesis" for the story. What this means, basically, is that seeing the "truth" of things requires insight and empathy, and that emotional truth is more essential that factual truth. St Exupery neatly demonstrates the problem through the pilot's drawings 1 and 2. Drawing No 1. really does look like a hat, and our failure to interpret it correctly puts us in the same place as the other "adults" the pilot dismisses. He finally spells it out for us, with drawing no. 2: what looks like a hat is really a snake swallowing an elephant. As readers, we are put in an uncomfortable position: we sympathize with the Pilot over the adults' lack of insight, but we (secretly) also feel a bit guilty ourselves, since we thought it was a hat too!
The Prince's problem -- how to understand the Rose -- is another case where the outside (how the Rose acts) is not the same as how the Rose really feels. The Prince learns that his ability to understand the Rose is connected to his experience of her. Although the Rose has told the Prince that she is unique in all the universe, on Earth he finds thousands of flowers that look just like her. Judging from appearances, the Rose would seem to be a liar, but what the Prince realizes is that none of them are her, because of the time he has spent with his Rose. In this way we can see that the truth of things is based on shared experience rather than what we see.