Why a spinning top stands up
Angular momentum, and how rotation resists being tipped over.
A spinning top stands up because its own rotation resists any push that tries to tip its axis over, converting what would otherwise be a simple fall into a slow, controlled wobble instead, and the faster it spins the more strongly it resists that push.
A battling-top arena raised a question nobody could answer, which was why a spinning top stands up at all, spinning fast and true on its narrow point while a stationary one of exactly the same shape falls straight over the moment it is merely nudged.
The physics of angular momentum
A spinning top has angular momentum, a quantity that combines how fast it is turning with how its mass is arranged around its spin axis, and that quantity points in a fixed direction in space, along the top's own axis, for as long as nothing interferes with it. A stationary top has no angular momentum at all, so the slightest nudge, the smallest imbalance in how it is standing, tips it straight over because nothing is resisting that tip. A spinning top has a real, physical quantity pointing along its axis, and gravity trying to pull that axis over has to fight against that quantity rather than simply having its way with an object that offers no resistance at all. The faster the top spins, the larger that angular momentum, and the harder gravity has to work to shift the direction it points in, which is why a fast top merely wobbles gently under a push that would instantly topple the same top standing still.
The frisbee comparison
Throwing a disc with a firm spin sends it gliding flat and level through the air, holding its tilt steady even as it curves and drifts sideways on the wind, because the spin gives it the same kind of angular momentum a top has, resisting anything that tries to tip its flat face away from the direction it started in. Throwing the same disc with no spin at all, a flat toss with just the flick of the wrist removed, sends it tumbling end over end almost immediately, because nothing is holding its orientation steady and the smallest unevenness in the throw is free to tip it over completely. The spin is not decoration on a good throw, it is the entire reason the disc holds its attitude, in exactly the way a top's spin is the entire reason it holds itself upright rather than toppling the instant it is set down.
The one number worth remembering
Doubling how fast a top spins does not simply double its resistance to tipping, since angular momentum grows in direct proportion to spin speed, but the wobble a given tipping push produces shrinks in that same proportion, so a top spun twice as fast wobbles at only about half the angle for the same nudge, which is why the same top looks rock steady moments after being spun hard and increasingly loose and wandering as friction slows it down over the following minute.
What follows from this
Because the steadiness comes entirely from the spin itself and not from anything about the top's shape once it has slowed, every spinning top eventually reaches a speed too low to resist gravity's pull any longer, and the same object that stood rigidly upright a few seconds earlier begins to wobble in a widening circle and topples, not because anything about it changed, only because the angular momentum holding it up had nowhere left to go but down as friction quietly spent it. A heavier top, or one with its weight pushed further from the point it spins on, carries more angular momentum for the same spin speed and stays upright noticeably longer than a light one given an identical flick, which is exactly why a good battling top is built with as much weight as possible sitting as far from the centre as the shape allows.