At about 80 million people, Egypt has about as many people as the rest of the Arab nations combined, and by far the strongest military in the region. Leaving it out of an "Arab League" is just plain unsustainable.
According to Syria and the Middle East Peace Process, the sticking point on getting Egypt back was Syria. Particularly Asad.
In the wake of the Camp David accords, all the other Arab nations both voted Egypt out, and suspended diplomatic relations with the country. Slowly over the next decade all except Syria restored diplomatic relations.
But getting Egypt back in the League required a majority vote, so it was a bit trickier. Assad's Syria was still unreconciled to Camp David, and fought hard against anything that might be seen as acceptance of it. Iraq and Jordan on the other side pushed very hard, finally setting up a rival organization, and pledging to boycott AL meetings while Egypt was not invited. Syria gave in, and Egypt was allowed back 3 months later.