Various meanings; see below.
Many people, especially those under 50 years of age, are clear what 'spitting feathers' means to them, that is, 'angry, or agitated'. Some also attempt to explain the derivation of that meaning as arising from the imagery of someone foaming at the mouth when angry, or even of losing feathers like an agitated bird in a cockfight.
Others, predominantly from the older generation, are just as clear in their understanding that 'spitting feathers' means 'extremely thirsty'. The imagery there is a little more intuitive and matches the American expression 'having a mouthful of feathers' and 'spitting cotton', and the Australian 'spitting chips' (of wood), which all mean extremely thirsty. That interpretation seems to be especially commonplace in the northern English counties, although it isn't limited to there. I can recall my father using it with that meaning, in the UK's Black Country region in the 1950s and, as anyone from there will be very ready to point out, the Black Country isn't in the north. In more recent years it has crept south and, in 1997, it cropped up in the BBC comedy series that was undoubtedly southern, The Fast Show:
"All joking aside, love, I'm spitting feathers here, let's have a nice cuppa."
How then do there come to be two meanings, with no apparent connection between them, for the same phrase? The 'thirsty' meaning has certainly had had the longer life. 'Spitting feathers' appears to derive from the earlier term 'spitting white' (to eject frothy-white sputum from a dry mouth). Shakespeare uses the latter in Henry IV, Part 2, 1597:
Falstaff: ... I brandish any thing but a bottle, I would I might never spit white again.
The invaluable Notes and Queries (A Medium of Inter-communication for Literary Men and General Readers), for January 1865, makes a link between the two terms:
"Surely Falstaff's spitting white is what, in Lancashire low life, is called 'spitting feathers'. A man, who has been drinking is feverish, his mouth is dry, and his saliva white."
The 'angry' meaning of the term doesn't make an appearance until the 1970s; for example, this piece from Davidyne Mayleas's novel Rewedded Bliss, 1977:
"Sometimes I could spit feathers. Tom and I have been married three years and he will still sometimes call me Tina. That's the name of his former wife."
The 'angry' interpretation of 'spitting feathers' is likely to have originated from another expression for angry - 'spitting blood'. Anyone who knew the latter term and heard 'spitting feathers' for the first time might very well assume that the two expression meant the same thing.