Any Rand was one of the most influential libertarians in the 20th century. She remained attached to the idea of the State because of her belief in the justice of retaliation. She called her philosophy Objectivism, because she believed in objective truth, objective values, objective justice, and objective control of retaliation. She defined government as follows:
A government is the means of placing the retaliatory use of physical force under objective control—i.e., under objectively defined laws.1
This explains what it is about government that appealed to Ayn Rand. If you believe in retaliation, the only alternative to a State which (ideally) retaliates against people in accordance with laws that are written down and enforced equally on everyone, is a system with competing retaliation agencies. These agencies would retaliate against criminals in different ways, which would be unfair. If retaliation were permitted in the absence of State penal laws, criminals would suffer unequal punishments for similar crimes, and some would suffer more for minor crimes than others would for major ones, depending upon the whim or arbitrary punishment theory held by the ones assessing the punishment. This was unacceptable to Ayn Rand because it is not objective enough.
Only a State, which enjoys a monopoly on the right of retaliation in a geographic area, can lend a sense of impartiality and uniformity to the administration of punishment and, by so doing, make retaliation seem objective.