'august' originates from Latin, specifically the word 'augustus', where the root is linked to Latin 'augēre' / 'auctus' (to increase) and 'augustus' carried the sense 'venerable, consecrated'.

'augustus' (Latin) was used as an honorific/title meaning 'venerable' and entered English (via Medieval and Early Modern Latin and literary usage) as 'august' retaining the sense of dignity and grandeur.

Initially it referred to being 'consecrated' or 'venerable' (often as a title); over time it broadened to mean 'majestic' or 'dignified' in general usage, which is the basis for the adverb 'augustly'.