I would like to ask you how to play like Cliff did it in 0:24. I can't understand how to do it.

Hammer-ons and pull-offs. Listen to his left hand, but watch his right.
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GK 1001RB-II
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^ What they said.

For the record: do not imitate Cliff Burton's technique. If he had lived longer (RIP) then he would have been facing severe tendon problems with his wrist curled and clamped onto the bass body like that.
Nice to see Cliff with something besides his usual Aria bass. R.I.P. brother.
"Maybe this world is another planet's hell?" - Aldous Huxley