Mario's incredible jumping abilities have been well-known for sometime now. For a stocky Italian plumber, the guy can get some serious hangtime - even a short hop is roughly as tall as he is. Nintendo's never offered an official explanation, because...well, it's Mario, and it doesn't really need an explanation.
That being said, fans have always loved coming up with theories about Nintendo characters (just look at The Legend of Zelda's timeline), and Mario is no different.
For Mario, things tend to be a bit simpler: either he's got some outrageous leg muscles, or gravity in the Mushroom Kingdom isn't quite as intense as it is here on Earth. So, what's the reason behind Mario's ridiculous jump height? Well, it's a bit more complicated than most people would expect. Here's how astrophysicist Gabe Perez-Giz explains it for PBS's Space Time:
So, not only is Mario's movement not limited by gravity, but according to science, gravity on Super Mario World is much more intense than anything on Earth. That's probably not the explanation people were expecting - not by a long shot.
When it comes down to science, Mario would have legs like the Incredible Hulk: with the Mushroom Kingdom's gravity being more intense than ours, Mario's legs would have adapted into something that could deal with the increased forces. Instead of regular ol' human legs, Mario would have some of the bulkiest and muscular legs of any living creature.
Of course, more and more problems begin to surface when logic is applied to Mario games (wouldn't his legs break on impact?), but it's strange that Mario's moon-jumping isn't based on gravity at all. As stated in the video, Mario would take off like a rocket if he were down here on Earth.
Then again, a Mario game in which the plumber is leaping over entire buildings on Earth does sound pretty amazing...