The Saudis rejected bin Laden's offer largely because it was arrogant and incredibly naïve of him to believe that the mujahideen, whose experience in Afghanistan was waging asymmetric warfare in a geographical setting perfectly suited to those tactics, could usefully apply those skills against a large, mechanised Iraqi army in the flat, open landscapes of Iraq and Kuwait.
The Saudi army was lavishly equipped, but relatively tiny; they had fewer than 60,000 troops, against the million-strong Iraqi army (then the fourth-largest in the world.) Saudi Arabia found itself with relatively few allies in the region; despite the objections of senior members of the House of Saud, including then-Crown Prince Abdullah bin Abdulaziz Al Saud, the United States was the most obvious — arguably the only — place to turn.
There's a wider context, however, to the always-difficult relationship between bin Laden and the Saudi regime. By 1990, the relationship was already strained, almost to breaking point: less than two years later, he would be banished from the Kingdom; in four years, he would be stripped of his Saudi citizenship.
But bin Laden nonetheless had connections at the highest levels of the regime: he had forged a working relationship with Prince Turki bin Faisal Al Saud, the long-serving chief of Saudi intelligence, the Mukhabarat. Prince Turki had been a crucial liaison between the CIA, the Pakistani ISI and the mujahideen. The interior minister, Prince Naif bin Abdulaziz Al Saud, had also been intimately involved with bin Laden during the Afghan jihad. The official to whom bin Laden made his offer of assistance in fighting Saddam was the defence minister, Prince Sultan bin Abdulaziz Al Saud.
But by the time of the invasion of Kuwait, there was already a breach emerging between bin Laden and the Saudi government. As bin Laden became more radical and more activist, both within the Kingdom and abroad, the government found him increasingly difficult to handle. His meddling in Yemen during unification in 1989-90, for example, infuriated the Saudis — in financing Al-Qaeda operations in the new Republic of Yemen, he antagonised powerful members of the Saudi royal family, from Prince Turki to Prince Naif to King Fahd himself.
Conversely, bin Laden became increasingly contemptuous of the government, which he came to view as deeply corrupt, insufficiently pious and in thrall to infidels. The decision to allow American forces into the Kingdom was an outrage bin Laden could not abide. By the mid-1990s, when the Saudis finally revoked his citizenship, bin Laden was deeply convinced that they were unworthy guardians of the two holy cities.
Good sources on the life of bin Laden and the evolution of Al-Qaeda are Lawrence Wright's The Looming Tower, Peter Bergen's The Longest War, and Steve Coll's Ghost Wars and The Bin Ladens.