The fool doesn't know what the wise man knows. But he knows what the wise man looks like. So in order to get the respect that the wise man gets, he takes on the wise man's appearance. He gets pictures taken and strikes poses and repeats what the wise man says so he may receive the respect and goodwill that the wise man receives. What the fool lacks, obviously, is wisdom - the thing that makes the wise man wise. And because he is a fool, he does not know this. So he comes to the foolish conclusion that he is now the wise man's equal.
But of course he is not. The only people the fool fools with his foolish pretensions is fools. And over time, the fool manages to fool himself too, and starts telling people that what he says is wisdom. But with time, fewer and fewer people are convinced of this and begin to question his alleged wisdom. And when the fool finds he is no longer being accorded the kind of wisdom people reserve for wise men, he accuses people of being fools.
What the wise man would have responded to with a shrug, the fool takes umbrage over. And in doing so, removes all doubt in the minds of those that surround him. They know now that he is, and has always been, a fool. They return to the wise man and apologise for thinking the fool wise. The wise man tells them that they are now wise. For no small part of wisdom is knowing the difference between true and false.