Batman mode is a toggle in the web client that forces all tiles to be colored black, all pathways to be colored white and all foreground color (items, tile types) to be colored white so things can easily be seen.
To activate batman mode use alt+5 or the 5 key on a numpad. It is a toggle so it must be used again to deactivate it.
Movement is accomplished via the 8 cardinal directions:
The web client binds the arrow keys and the numpad keys to their appropriate direction. Each direction also has a shortened alias of the first (or relevant 2) letter of the direction.
Pathways are links from one tile to another on the same or a different map. Pathways have a destination and a name. To use a pathway utilize the Enter command while standing on the tile of a pathway with the syntax:
Enter [pathway name]
Pathways are visually denoted by a colored dotted border around a tile. There may be multiple destinations per pathway so check the pathway by mousing over the tile for the right name.
NOTE: Pathways may not always be easily seen due to color conflicts. You can use Batman Mode to see any potentially hidden pathways. See the help for Batman Mode for more information.