This type of visa is created by US government to allow the O-1 and O-2 visa holders to bring their children or families along with them during their work stay in the country.
It is important to note that only spouse or unmarried children under 21 of O-1 and O-2 visa holders will be granted such visas. No other person, such as parents or close relatives, will not be eligible for an O-3 visa.
To apply for an O-3 visa, you can apply along with the O-1 or O-2 visa holder or after the principal person has secured their O-1 or O-2 visa at a US Embassy where you live.