Because pronouns (ex. his, its, their) take the place of nouns, they must mean the same thing as the nouns they replace. Here, to decide which noun is being replaced, ask, "Whose affairs (business) did the British lack the capacity to...intervene in"?
The British could not intervene in their own affairs, so it must have been the "continent's." "Continent's" is singular, possessive and not a person, so "its" is the appropriate pronoun. ("It's" is not possessive, but a contraction for "it is.")