Why were the conquered people accepting of Islam?

Because they discovered it is the right thing to do.

Mongols tried to invade the whole world, Europe was saved when they stopped their invasion to return to their capital to select a new Khan.

Mongols faced Muslims i the battle of Ain Jalut in palestine where they suffered the first defeat in their history.

Later some of their kings accepted Islam after comming in contact with it and understanding the great values it has.