No explanation is ever given in canon.
One popular explanation, however, relies on the act that the hero (He-Man, in this case) puts up as the civilian (Prince Adam). In the minds of those around him, He-Man is defined by strength, both in the literal sense and in the sense of his character. He's reliable, he plans carefully (and knows when he needs to consult better minds than his), he's brave, he's strong and fit, and so on.
Prince Adam isn't evil, but in the minds of those around him, he is defined by weakness. He's irresponsible and careless. Cringer helps the act along, by playing at weakness in areas that Adam cannot: he's notoriously cowardly and lazy, and because he is Adam's constant companion, Adam becomes associated with those weaknesses even though he doesn't necessarily have them himself.
In the face of such contrast, it becomes hard to believe that these people are one and the same. Of course they look similar, but come on; He-Man is strong. Prince Adam is weak. Everyone knows you shouldn't judge a book by its cover, and you only have to look a couple of pages in to see how different they are, so why bother finishing Adam's book? It's not very likeable anyway.
And that's exactly what the act is designed to make people think. In some ways, you could argue that this subverts some of the lessons that the story aims to teach, because if people tried to judge them by looking, it would be clear that they were the same person. Or you could argue that this illustrates the need to look deeper: sure, people could stumble across the truth just by looking, but that's only a coincidence. People don't figure out that Adam is He-Man because they don't like Adam, and so even if they look past the surface, they stop looking before getting very far past it. And that's why they wind up not seeing the truth.