Relax in a slothful and mindless manner.
This phrase dervies 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 in the 1990s. Most of the early uses of it come from the London literary elite, although the earliest I’ve found is in the 1990 film Pretty Woman. In that, Julia Roberts’ character uses this line:
“Let’s watch old movies all night… we’ll just veg out in front of the TV.”