Rightful claim on what exactly?

A claim here refers to a claim to the throne of Scotland. When a king dies, succession can be a messy affair if there are multiple people who claim the throne to be theirs. Any claimant will require support to enforce their claim, or else others will force them out. Here, Balliol requests support for his claim to the Scottish throne, so that he may rule as king, rather than the other people who claim the throne for themselves..