The mobile crane of course has its steel cable which is what the hook is suspended from. That runs from a spool on the cab/base with a motor and up the boom over a pulley connected to the hook.
The backside of the boom has those two "wings". Attached to them are two cables that are connected to the boom with a pulley. Those cables go down to the base of the crane with another motorized pulley. So when the crane is lifting a load, that motorized pulley pulls those cables taught. The tension keeps the load balanced so that it doesn't tip too much in one direction. The base/cab/truck of the crane also has outriggers giving it a larger platform for more stability.
The telescoping part of the boom is "telescoped" or shortened for transport. The piston part of the crane boom can be shortened to the length of the crane truck so that the crane be driven on public streets. The lattice part of the crane on the very end is removed and transported on a flatbed truck. It assembles pretty much the same way a tower crane does using huge steel pins to keep both pieces together. The lattice section is attached using another mobile crane.
Some of these mobile cranes are able to lift up to 130 tons. Even when they're lifting tower crane pieces like this, they're likely only lifting a fraction of what they can lift. Most city buses for instance weigh between 11 and 14 tons.
If you check out the JW Marriott construction cam, you can see the construction/deconstruction of the mobile crane. They took it apart on July 7.
http://oxblue.com/open/tournee/jwmarriottaustin