According to Word Detective, go spare originally meant be made redundant, and the anger was a secondary effect:-
The original sense of “go spare,” when it first appeared in British
slang in the 1940s, was “to be or become unemployed,” making it a
close cousin of the more formal British euphemism for being laid off,
“to be made redundant.” By the late 1950s, the normal emotional
reaction to losing one’s job had colored the term “go spare,” and it
had had acquired the added meaning of “to become distraught or very
angry” (“When he saw what I had done he went spare,” 1958).
I can't say that I find this explanation particularly convincing, but I offer it for what it's worth.