What "brilliant" means
A brilliant move (the classic !!) is a strong move that is also hard to see — most often a sacrifice. You give up material, and the engine confirms that the position that follows is winning or clearly better anyway. The combination of "gives up material" and "is still the best idea" is what earns the double-exclam.
Why it is not just the best move
A move that keeps a big advantage is only "best", not brilliant. Brilliance requires a twist: the natural-looking move would be to hold onto your pieces, and instead the winning path is to let one go. If keeping the material was also fine, the sacrifice is not truly needed — and most review engines will not award !! when the give-up is the only move, or when it is a simple, even trade.
How a game review detects it
When you run a game review, the engine checks two things at each move: did you give up material (using a static exchange evaluation, not just piece count), and is the resulting position still winning? If both are true and the move is not forced, it is flagged brilliant. That is why brilliancies cluster around tactics — a queen sac into mate, a piece for a decisive attack, a quiet move that wins despite hanging a rook.
How to find more of them
Look at your opponent's undefended pieces and your own forcing moves before you settle for the safe recapture. Then review the game for free afterwards: seeing where a sacrifice would have won trains your eye to spot the next one. If you want to know how the other labels are decided, read blunder vs mistake vs inaccuracy.