ELI 5: Have you ever wanted to hit your annoying brother but all you have in your hand are a few sheets of paper? If you want to inflict a respectable level of hurt, you have to roll them up! Curved paper doesn't bend as easily as flat paper. With a spinning top, it's the same idea, but in one higher dimension (time). Spinning an object is like curving its 4-dimensional shape. It won't "bend" (fall over) as easily.

ELI 15: Mathematicians say that the "geometry" of an object doesn't change as long as the distance between any pair of points doesn't change. When you roll a piece of paper, the distances (on the paper, not through the air) don't change, so any properties of the object that only depend on those distances will not change. Gauss famously derived such a property/quantity for 2-dimensional shapes (like a sheet of paper), which we now call "Gaussian curvature." Riemann later generalized this to higher dimensions, and Einstein later showed that gravity is a manifestation of the "intrinsic curvature" of spacetime by showing that the familiar concepts of energy and momentum are properly understood as values derived from Riemann's "curvature tensor."

When you spin your gyroscope, you aren't changing it's "geometry" because it isn't stretching or ripping. Therefore, its "Riemannian curvature" isn't changing either. In the same way that a sheet of paper maintains its intrinsic curvature by not bending once it has been rolled, a gyroscope maintains its (4-dimensional) intrinsic curvature by not falling over when spun. Similarly, a spinning fastball travels straight, while a non-spinning knuckleball moves erratically. The knuckleball is the higher-dimensional analog of a floppy piece of paper.

What I find appealing about this explanation is that it's based purely on the mathematics, and it doesn't make any reference to the mysterious "physical law" called the "conservation of angular momentum." I've never heard a physicist tell me why angular momentum is conserved. "It just is."