Meanings
To defend a cause or support ideas, with more effort than the prevailing authority. It could also indicate intransigence and intolerance.

Examples
In this country the King is held in a certain position which is inviolable and the people will never ever tolerate criticism of the King: sometimes I wonder that they are more Catholic than the Pope, but they are more royalist than the King for sure
In questa nazione il re è considerate inviolabile e la popolazione non permetterà mai che il re venga criticato: a volte mi chiedo se la gente qui sia più cattolica del Papa; di sicuro è più realista del re

Origin
Description of the Ultras group (plus royalistes que le roi), which were the dominant political faction in France under Louis XVIII (1815-1824) and Charles X (1824-1830)