Why did Verbal get legal immunity?

I think the real reason Verbal got immunity is because he (as Keyser Sose) has bribed the district attorney and/or other people with the power to grant the immunity, to facilitate Verbal's release from custody.

The cop Kujan references this when he says "Someone is out there pulling strings for you" and "Captain Leo wants you out of here ASAP." At an earlier point Kujan had also angrily mentioned "that precious immunity they seem so fit to grant you."

Kujan seems to recognize the immunity is bizarre, perhaps because immunity is usually given to a small-time criminal in exchange for ratting on a bigger criminal -- but in this case (as mentioned in the first clip above) Verbal Kint does not even have to "turn state's" which means he does not even have to testify to anything -- so the cops are getting no value out of the immunity.

Kujan misinterprets the bizarre immunity as someone freeing Kint just to kill him later (see first clip above). Of course, at the end of the film, Kujan realizes his mistake.

Kujan did get a long, off-the-record backstory of the crime out of Kint, but not because Kint was exchanging the backstory for immunity -- Kint only tells the backstory after Kujan threatens to ruin Kint's reputation on the street (see second clip above). And the backstory, of course, turns out to be huge misdirection, probably spun by Kint on the spot, just to make a fool out of Kujan for trying to get tough.

Your question contains the answer within its own contradiction;

Why did Verbal get immunity when he was clearly involved in the crime?

If he hadn't been involved in the crime, why would he need immunity in the first place? an innocent man has nothing to gain from immunity.

Granting immunity from prosecution for an individual is perfectly legal, and enacted typically when the full co-operation of said individual will facilitate the release of wider evidence, or insight. In the instance of The Usual Suspects, Verbal is the only witness, as everyone has died. For the authorities to find out what has happened, they have to secure the co-operation of Kint; and they're going to have a hard time doing that if they lock him up.

As for the cash van... was that ever really 'Keyser Söze's motive?

Verbal got immunity on the grounds of helping investigators in solving the mystery of freighter ship.

Source wiki:

Verbal has beengranted immunity from prosecution provided he assists investigators, including Customs Agent David Kujan (Chazz Palminteri) and reveals all details of his involvement with a group of career criminals who are assumed to be responsible for the destruction of a freighter ship and the murder of nearly everyone on board.

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