The thirteen colonies were primarily obtained by Britain through a combination of methods, including treaties and settlements. However, the best fit among the options you've provided would be:
by settling colonists on claims made by explorers.
Initially, colonies were established by English explorers and settlers who claimed land in North America. Over time, through a mix of charters, land grants, and colonial expansion, these settlements grew into the Thirteen Colonies. Additionally, some colonies were acquired through military actions against the Dutch, but the predominant way the colonies were established was through exploration and settlement.
If we consider the role of treaties, like the Treaty of Paris (1763), it did play a part in solidifying British control after the French and Indian War, but it primarily impacted territories beyond the Thirteen Colonies rather than the colonies themselves.