I have tried googling this, but I still only find answers that tell me this property exists, and not an intuitive explanation as to why.
4 Answers

The starting point is basically inertia and the end point is conservation of angular momentum.
Inertia – staying at the same (linear) velocity and resisting change in velocity – is a general principle. But we can think of mass as the quantified measure of inertia, ie exactly how much a given body resists changing its velocity when a force is applied. The measure being the ratio of an applied force to the change in velocity, ie m = F/a defines inertial mass.
The next step was the mathematical discovery that a certain quantity (mass x velocity) was often conserved in ideal circumstances. This quantity can be loosely considered as “the amount of moving going on” if you consider a unit mass at a unit velocity as being a unit of moving. Formally it is called momentum (although Newton just called it motion).
The conservation of linear momentum can be extended into rotating bodies as angular momentum. And that is what happens with gyroscopes. I have never been able to intuitively understand the way they move when a force is applied to the axis but my general understanding is that it moves like that to conserve angular momentum in all planes of a 3D space.

The answer is : ” spin angular momentum”

You won t get a simple, intuitive explanation because the process by which a gyroscope works is not a direct application of intuitive experience.
However it can be understood if you have a grasp of some underlying physics and are willing to take some time and think hard and carefully.
There are many explanations available. The one in the video (link below) is quite good.

Good description here