I'm sad that every explanation of MCMC uses complicated symbolic explanations. I was trying to explain it once and came up with a fairly intuitive way to visualize it. Probably that is the way it was first discovered, though who knows.

In words, you can visualize it as doing a random walk around the area of a probability graph. Then it just turns into an explanation of why random walks accurately sample from an area, and the various tricks used in MCMC methods to make random walks more efficient.

Relatedly, bayes theorem makes much more sense when you visualize it, e.g. this post: https://oscarbonilla.com/2009/05/visualizing-bayes-theorem/