It was an attempt by Kaiba to take Grandpa's "Blue-Eyes White Dragon" card.
The Duel takes place off screen. It is based on their Duel in the manga. Although the events leading up to it are based on their first encounter.
Kaiba entered Kame Game and found that Solomon possessed a "Blue-Eyes White Dragon" card. Kaiba already having three copies of the card, wanted to possess the fourth so he would be the only person in the world with the card. However Solomon refused to trade the card or sell it, because of its sentimental value to him.
Kaiba later had Grandpa kidnapped and brought to KaibaCorp. Kaiba proceeded to beat him a Duel and called Solomon's grandson, Yugi, to come and collect him. When Yugi and his friends arrived, Kaiba ripped up Solomon's "Blue-Eyes White Dragon" in front of them, saying it could not be used against him now. To get revenge Yami Yugi then faced Kaiba in a Duel using Solomon's Deck.
- Yu-Gi-Oh! episode 1: "The Heart of the Cards"