I know the vector equation that relates torque to moment of inertia and angular momentum. What I want to know is what physical mechanisms actually occur to keep the gyroscope from falling. Where is the upward force coming from and how, mechanistically, does this get produced?
Since you don't want a math description, but rather a physical intuition, I'll attempt to give one.
The opposing force comes from the fact that when you try to rotate the gyroscope, you are attempting to redirect the path of the material in the spinning ring. Imagining the spinning ring as a discretized collection of point particles in motion, you are essentially trying to push on these moving point particles to divert their path.
When you punch a baseball to divert its path, you feel a force against your hand which opposes your attempt to divert its path. The exact same physical/tactile intuition applies to the gyroscope, once you view the interior ring as a collection of moving point particles whose motion you're trying to divert.
A faster moving baseball requires more force to deflect its path by a certain angle; similarly, the faster a gyroscope ring rotates, the more effectively the assembly will resist your attempts to rotate it.