The highlighted dependent clause looks wrong to me, as I can't pinpoint the subject of "what did not appear"? "on what grounds" is a prepositional phrase functioning as an adverbial adjunct.
The sentence is perfectly grammatical. If you read the whole paragraph, Oliver filed with the court (presented) a petition for appointment of new trustees of the testator‘s will. This (referring to the petition) was opposed (resisted) by Ray & Bush, but the grounds for opposing the petition did not appear (were‘t stated in anything Ray & Bush filed with the court in opposition). Nonetheless, the petition was denied. So “on what grounds” refers back to the petition for appointment of the new trustees.
The petition was resisted but the grounds for resisting it were not provided.
A modern, more ordinary (but less pithy) phrasing would be "the grounds in support (of the resistance to the petition) were not presented to us (for consideration or scrutiny)".
"Grounds" is used in the sense of "a set of one or more reasons comprising a justification (of or for something)".
"Appear" is used in the sense of "move or arrive within view (of a particular person)".