It is hard to talk about without demonstrating. Its the same concept that makes you stay up on a bicycle. Imagine if you had a rock on a string and spun it around over your head. if that string breaks, the rock will shoot out in a direction in a straight line tangent (i.e. 90 degrees) from the point on the "circle" you are making from twirling it around. This gives you the momentum.
Since newtonian physics tells us that an object in motion will want to stay in motion unless a force is applied to change it, this causes a spinning mass to be stable around an axis. To change the axis requires force to change the direction that rock wants to go.
If you keep this spinning mass isolated from outside forces (instruments might use low friction ball bearings to do this), it will tend to stay stable while the outside area (the airplane) moves. So when you look at a gyroscopic instrument "move" you are seeing it stay still while you and the aircraft are moving around an axis.
Hope that helps.