Even though Kaiba is the wealthy CEO of his own multinational high-tech corporation KaibaCorp that invented the virtual reality duel system, his real passions lay in the world of Duel Monsters. He uses his ruthless business drive to overcome any who challenge him. Kaiba wants to be the greatest and most powerful duelist in history, and he knows that Yugi will be his eternal rival.
Season 1 (Seven Stars arc): It's revealed that the school was built over the hiding place of 3 powerful, Evil Counterparts of the Egyptian God cards and that its purpose is to raise students strong enough to defend them (the gate sealing the cards can only be opened if its 7 keys are won in a duel). Naturally, Jaden and co. are entrusted to guard said keys from the Hidden Agenda Villain's henchmen.
Lampshaded in Yu-Gi-Oh! The Movie: Pyramid of Light. After Anubis' defeat, Kaiba leaves in a huff, and Grandpa says, "I'm glad he didn't bring up the damage you all did to his Duel Dome, because I really don't think his insurance is gonna pay for it!"