What's the meaning of the phrase 'Veg out'?
To 'veg out' is to relax in a slothful and mindless manner.
What's the origin of the phrase 'Veg out'?
This phrase 'veg out' derives from the association of vegetables with mental incapacity; in the way that mentally disabled people are sometimes referred to as vegetables. 'Couch potato' comes from the same notion.
The term originated in the 1970s. Most of the early uses of it come from the USA, although the earliest that I know of is from Canada, in The Calgary Herald, November 1979:
...the teenager's favourite description of doing nothing. ("I'm just going to veg out today.")