The fate points for a player are on their character sheet. The player drags a point off their sheet to the chat frame to use a personal Fate point. This will add the comment to the chat window and decrements their Fate point total.
The Fate points on the desktop are for any group Fate point (if the GM wants to use that) and for GM Fate points (for NPC usage). Double-clicking adds a point to the pool. Dragging off decrements the available points.