Thor is not immortal, just superhumanly strong. He can still be hurt and die. Loki says:
The humans think us immortal. Should we test that?
He's talking about the human belief that Asgardians are the literal Norse Gods and have the immortality associated with it. In the MCU, the Asgardians are just sufficiently advanced aliens.
The reason Thor attempts to break out of the cage are twofold. One, he wants to get back to the fight as soon as possible. Loki's still messing around on the helicarrier. Two, falling straight down at terminal velocity from miles above the Earth will hurt.
Guessing Thor is 400 lbs (Super Dense Muscle), falling from a height of 35,000 ft, and going only 0.1m into the ground, would result in 162,196,272 newtons of energy. Converted into Pounds per Square Inch, that's 23,524 PSI in a fraction of a second. Yay physics. The human body at sea level experiences 14 PSI, and can survive up to 400 PSI if slowly increased. Sudden increases are deadly. An increase of 30 PSI in a fraction of a second will kill you. A standard US military hand grenade of 0.4 pounds of TNT worth of explosives provides 836,800 newtons of energy or 121 PSI at it's source.
That means the crash of the cage would have been extremely powerful. By breaking out and redirecting his momentum sideways, Thor would reduce the amount of energy relative to the Earth (like the International Space Station does resulting in microgravity "Zero G"). Notice that Thor is still thrown around wildly, but alive.
Keep in mind, the actual force of the impact would be alot higher, as the weight of the cage was not taken into consideration.