The New Partridge Dictionary of Slang and Unconventional English (edited by Tom Dalzell, Terry Victor) provides the first usage of the slang verb stack meaning 'to crash (a vehicle)' from 1971:
1 to crash a vehicle Australia
- He had lost his way. Stacked his Harley Davidson.
— Jack Hibberd, A Stretch of the Imagination, p. 8, 1971
The sense 'to fall' appears to be a semantic extension from the sense 'to crash' and some Australian dictionaries list a more detailed definition of this sense. For example, The Big Book of Australian Slang (by Steve-O Swagman) provides the definitions below for the extended senses:
Stack: To fall, often hilariously, either due to/while being legless on a wheeled vehicle.
Stack hat: A protective piece of headwear used to prevent sh-t riders of bikes, skateboards and other open-aired vehicles from stacking and suffering brain damage. A helmet.
Stack it: This phrase is used, often mockingly, to refer to someone who has lost their balance in an overwhelmingly comedic fashion. This can be used when the subject falls for no reason at all, being way too pissed, or attempting and failing a trick on a bike, skateboard or similar object.
I believe the verb stack is originated as an equivalent of pile (up) which also means 'to crash a vehicle' in colloquial usage with an earlier date of origin (from 1891 per OED), and the corresponding original sense of the verbs is same. The meaning in informal usage appears to be an analogy; as when you crash, things tend to pile (up), stack (up). Pile-up is used for a multi-vehicle crash also. Here is the equivalent original sense of the verbs stack and pile from OED:
pile transitive. To form into a pile or heap; to heap up. Also in extended use. Usually with up, on.
stack: transitive. To pile (corn, fodder, etc.) into a stack; to make a stack of, to pile (something) up in the form of a stack.
Additionally, stack and pile-up (pile) share the same meaning in American football and Australian rugby jargon also.
The Australian Oxford dictionary (published by Oxford University Press) lists the sense 'colloq. crash (a motor vehicle)' for stack also; and includes the colloquial phrase stacks on the mill meaning:
situation (in Australian Rules, Rugby, children’s games, etc.) where several players are piled up promiscuously, chaotically, one on top of the other (with the ball at the very bottom of the pile).
Note: I've also asked my Aussie friend and he had no idea of the origin of the slang word stack; but he confirmed that it is used in the current vernacular.