The Tile Rack window is intended to look like a rack for holding tiles from a Scrabble game. The author's experience in the real world (as you are probably already aware of) was to use these racks to hold a player's tiles (a major shortcoming in the packaging by Avalon Hill).
This window has room for 9 tiles (the 3 extra spaces are to allow you to move the tiles around and group them). The tiles are shown in the window as push-buttons.
When it is your turn to play, clicking any tile on the Tile Rack will cause that tile to be played. Clicking on a tile while not your turn will do nothing relevant to the game, the status bar will note that it is not your turn.
The Tile Rack has enough room to separate the tiles into up to 4 groups. This would typically be done to group tiles that: form chains, merge chains, grow chains, are not playable, etc.
To move tiles around on the Tile Rack, hold down the SHIFT key and drag the tile to any of the nine possible positions. If the tile is moved over or near a spot occupied by another tile, the 2 tiles' positions are swapped. Tiles cannot be moved off the tile rack.
Tiles on the Tile Rack are colored based on their playing status. The colors are automatically updated as the game progresses. Colors:
Grey - Indicates an open tile that does not abut any other tiles.
Light Green - Indicates tiles that have the potential to become formers. There is always 2 or more tiles that will be colored as such. If one is played, another may be able to form a chain at the player's next turn.
Chain Color - Indicates a tile which would increase the size of the indicated Hotel Chain.
White - Indicates a tile that would either form or merge Hotel Chains.
Dark Grey - Indicates a tile that is unplayable at this time. The caption for the tile will also have a strike-through. This is done for tiles that would form a Hotel Chain if there were any available to form.
Light Grey - The tile is a dead tile (it merges 2 safe Hotel Chains). The caption for the tile will also have a strike-through. Dead tiles are automatically replaced at the end of your turn.