Keyboard and Button Commands
Most menu items can be bound to joysticks buttons or keyboard shortcuts
LiveTraffic offers "commands" for the most important menu items. These "commands" show up in X-Planes configuration screens for Joystick and Keyboard asignments. All of them are grouped under the top-level category "LiveTraffic".
Menu Item
Aircraft Info...
Opens an aircraft information window
Aircraft Info... (Popped out)
Opens a popped out aircraft information window (separate OS-level window)
Aircraft Info shown
Hides/Shows all aircraft information windows, but does not close
Close All Windows
Closes all aircraft information windows
Aircraft displayed
Starts/Stops display of live aircraft
TCAS controlled
TCAS Control: Tries to take control over AI aircraft
Labels shown
Toggle display of labels in current view
In X-Plane, the description is shown as the text to select in the configuration dialogs.
All commands are executed on key/button down event only, i.e. just once, no matter how long you press a key or button.

Keyboard Assignment

Go to X-Plane's settings for Keyboard. See here for X-Plane's manual on using the settings dialog. You find the LiveTraffic commands most easily when selecting Plugin Provided:
LiveTraffic's commands in X-Plane's Keyboard settings
Assigned keyboard shortcuts appear on the menu next to the respective menu item, which helps remembering them.

Joystick Button Assignment

Similarly, when assigning joystick buttons (see here for X-Plane manual) you can refer to the very same commands in the right hand command selection:
LiveTraffic's commands in X-Plane's Joystick settings