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This joke plays on the massive gap between "everyday" visualization and one of the most famous, unsolved headaches in physics.
To get why this is funny (and a bit self-deprecating), you have to look at the two different ways the phrase "3-body problem" is being used:
1. The Scientific "3-Body Problem"
In physics, the Three-Body Problem is notoriously difficult. While we can easily calculate the orbits of two objects (like the Earth and the Sun) using F = G \frac{m_1 m_2}{r^2}, adding a third massive object makes the math chaotic.
There is no general "closed-form" solution that can predict exactly where those three objects will be forever; the system becomes unpredictable over long periods. Solving or even visualizing the complex gravitational interactions of three bodies is a feat of high-level mathematics.
2. The Punchline: Literal vs. Mathematical
* The Set-up: The first friend is talking about Stephen Hawking, who was famous for his ability to mentally manipulate 11-dimensional M-theory and black hole singularities—stuff that is literally impossible for the average person to "see" in their head.
* The Twist: The second friend hears "3-body problem" and thinks it just means "having three objects in your head at once."
* The Irony: He is "proudly" claiming he can visualize the Sun, Earth, and Moon. To a normal person, that’s just a basic mental picture of space. To a physicist, "visualizing" a 3-body problem means calculating the chaotic gravitational dance between them.
The Verdict: The friend thinks he’s a genius because he can picture three things at once, whereas Hawking was "visualizing" the actual underlying physics of the universe. It’s like someone saying, "Oh, you're a master chef? Well, I once made toast, and that's kind of a culinary challenge."

You visualize n-dimensional space and then set n to 4.