This is how I understand it:
While the aircraft is sitting there on the ground, the east-west gyro is sensing the rotation of the earth. Obviously all the bits of the earth's surface go around the earth's axis once per day, but the bits at the equatorial latitudes have further to go, so they are going faster. The gyro senses how fast left to right movement seems to be. It then compares that to the amount of west-east movement that there should be at that latitude. (It knows the latitude because you just entered it). Now it's just like a crosswind calculation: if the crosswind ("crossearth"?) component is so much, and the total wind (earthrate) is so much, then what angle must you be at to the wind (earth's rotation)? The earth rotates about the axis that points to true north, so this process has determined the position of true north.
Atomic has provided the last piece of the jigsaw. Given true north, the computer consults the World Magnetic Model to determine magnetic north in relation to true north.
On another thread, someone whom I would credit if I could find the thread, pointed out that north of 70N (or south of 70S) the speed of rotation isn't sufficient for accuracy on the IRS setting, so the system is coupled to a GPS, which presumably can tell its orientation to the satellites it has found.
I believe the task of sensing the Earth's rotation accurately is the one that takes the ten to fifteen minute start up period.