Briefly means for a short time, or in few words. The sun came out briefly between two showers; 'tell me briefly what happened' is a request for the short version.
Which of the two you get comes from the sentence around it. For time: she paused briefly, he was briefly famous, the shop closed briefly for repairs. For words: 'briefly, the plan is this' at the front of a sentence promises a short answer, and 'to put it briefly' does the same. The family around it is worth having — brief as an adjective (a brief meeting), as a noun (the brief you are given before a job), and as a verb (brief the team), with brevity for shortness itself. Brief is about time or length, never size: a brief man makes no sense.
Briefly is brief with -ly on the end, and Middle English took a while to settle the spelling — breefly, breffly, brevely, breveliche and brevelike all sat side by side before one of them won.
Middle English
breefly
English, today
briefly
From Middle English breefly, breffly, brevely, brevelike, breveliche, equivalent to brief + -ly.
Etymology from Wiktionary, CC BY-SA