That means if you apply force to change the direction of their axis of rotation, the axis actually moves at right angles to that force.
The faster they are spinning, the more force it takes.
A simple example is the front wheel of a bicycle or motorcycle.
If you are traveling at speed, and you press forward on the right handlebar (as if you wanted to turn the wheel to the left)
what happens is the wheel doesn't turn in the direction you are pushing it; instead it precesses and tilts you to the right.
So to initiate a right-hand turn, act as if you want to turn left!
As another example, the WW1 aircraft, the Sopwith Camel, had a rotary engine. In other words, the entire cylinder block rotated, making a powerful gyroscope.
So pitch inputs would result in strong yaw precession, and vice-versa, killing inexperienced pilots.
So if you want to stabilize a walking platform with a gyroscope, keep in mind that it doesn't always point in the same direction.
You have to provide right-angle corrective forces to make it point in the direction you want.
ADDED: You might be wondering why car wheels don't do the same thing. The answer is that they want to but can't.
So if you press the steering wheel to the left, the wheels want to precess so they would be tilted to the right.
Since they are constrained so they can't tilt to the right, this generates a force trying to tilt them upright.
That force causes the wheel to precess in the direction you originally tried to turn it!
So it acts as if you can easily turn it to the left, but there are precession forces being born by the wheel bearings.
NOT TO BELABOR, but there's a very simple way to understand precession.
Don't think of the gyroscope as a solid wheel.
Think of it as a bunch of independent weights connected to the hub by strings.
So imagine taking your bicycle and elevating it off the ground, and spinning the front wheel (in the normal direction), thinking of it as a bunch of independent weights spinning.
Applying a force to try to turn it to the left is like placing your hand against the stream of weights at the front of the wheel (where they are descending) and trying to push them to the left.
Rather than moving to the left, they
bounce off your hand into a slanted direction down and to the left.
So then they are orbitting in a slanted direction.