It's a common misconception that a gyroscope resists having its axis rotated. More accurately, it responds paradoxically to such forces by moving at almost right angles to them.
Imagine trying to push a shopping trolley with its wheels jammed way to the left. If you try to push it forwards it goes left. If you try to resist its going left, you will feel it resists moving forwards, but that's your own resistance played back to you.
It's much the same with a gyroscope. If you mount it in a gimbal so that it is free to turn either way, then rotate one part of the gimbal manually, the gyroscope responds by rotating in the other swivel direction. You won't feel much resistance now.
As to why... imagine a wheel rotating, top away from you. Suppose you wish to tilt the wheel to the right. The part nearest you needs to switch from moving vertically upwards to moving up and to your right. That's a change of velocity, requiring a centripetal force to your right. Meanwhile the part furthest from you needs to switch from straight down to down-and-to-your-left. That requires a force to your left. So the torque needed is to the right near you and to the left far from you, which is around a vertical axis, not around the horizontal axis as you expected.