Gyroscopic precession is a mysterious force, until you look at the fundamental physics behind it. The best way to figure it out is to look it up in a Physics text, or an Engineering Mechanics text.
The underlying principles are basic. The gyro has a fixed angular momentum due to the spinning mass. This is a set quantity, and(like linear momentum) must be preserved. If you try to rotate the gyro, you are adding to the angular momentum (because the rotation of the whole gyro in another plane induces a change to the total angular momentum). The faster the rotation you impart, the more the momentum change that you request. As a total system, the momentum must be constant, when you disturb the angular momentum balance, the extra rotation 90 degrees out results.
One way of describing it is to picture a wheel spinning free in space. Imagine it in front of you spinning with its axle pointing at you, rotating clockwise (top to the right). It is happy to spin there, undisturbed, until you push on the top rim (12 o'clock position)in a direction away from you. The wheel is rigid, of course, so each part of it cannot move relative to any other. Because you start it moving away from you, the piece of the wheel at the top has two components of its velocity, one to the right, and one directly away from you. This means that you have tried to disturb its velocity, relative to the rest of the wheel. Because the rototion you called for (top away, bottom toward you) has no rotational speed at the sides of the wheel (the 3 and 9 o'clock pieces of the wheel see no translation due to your push), the mass has the confusing need to have more energy at the top and bottom (more total speed) than it does at the sides (3 and 9 o'clock) The "precession" of the gyro is simply the motion that the gyro makes that rotates this 3 and 9 o'clock part to EXACTLY match the 12 and 6 o'clock part. The speed of rotation imparted on the wheel is proportional to the force you exert on the wheel, and its angular momentum (speed of rotation, basically). The slower it spins, the faster the precessional speed to make up for the larger disturbance you caused.
Try this site for one stab at the explanation:
http://sprott.physics.wisc.edu/demobook/chapter1.htm