Tolkien and Lewis started it when Tolkien returned to Oxford.
I'll answer the questions that you mentioned one by one. Unless otherwise specified, the quotes come from Inside The Lion, The Witch, and The Wardrobe by James Stuart Bell, Carrie Pyykkonen, and Linda Washington, chapter 5.
When was it set up?
Years later (after working on the Oxford English Dictionary), when he returned to Oxford as a professor of Anglo-Saxon (Old English), Tolkien and Lewis founded a writer's group called the Inklings.
According to Wikipedia, this was in 1925. So the earliest possible year is 1925. But the Wikipedia article on C. S. Lewis claims that they met for the first time in 1926, pushing the possible date farther forwards. In doing some Internet sleuthing, I looked at the source that Wikipedia provided for when they ran - from a book called Brothers and Friends: The Diaries of Major Warrington Hamilton Lewis, C. S. Lewis's brother. I then looked up the book on Google Books and performed a search for 'Inklings'. On page 183, there is this:
-ings of the Inklings did not begin until April 1940. Though members of the Inklings often met at various times throughout the week, the regularity of Tuesday mornings
So maybe the Inklings were founded in April 1940, although that could be referring to when they started regularly meeting.
Update: Doing some searching through Google Books, I found this from The Inklings by Humphrey Carpenter:
These three men knew each other well. Lewis and Tolkien met in 1926 and soon achieved an intimacy which lasted for many years. Around them gathered a group of friends, many of them Oxford dons, who referred to themselves informally and half jestingly as 'The Inklings'. When in 1939 Charles Williams found himself obliged to
So apparently it just sort of developed after Tolkien and Lewis met.
This group met weekly (sometimes on Monday or Tuesday mornings, other times on Thursday evenings) during the 1930s and '40s...
Who were its founding members?
See my first quote - it looks like Tolkien and Lewis set it up themselves.
...1930s and' 40s and consisted of Lewis, Tolkien, and Charles Williams: Owed Barfield (Lewis was his daughter's godfather); Warren Lewis; Nevill Coghill (an Oxford professor); John Wain (not to be confused with John Wayne, the American actor famous for westerns - this is John Wain, the English poet and novelist); Gervase Matthews (a lecturer at Oxford); Hugo Dyson (whose talk with Lewis along with Tolkien helped Lewis to believe in God once again); Robert "Humphrey" Havard (another lecturer, who was also Lewis's and Warren's doctor); Lord David Cecil (an Oxford professor); and others, including Tolkien's son Christopher. Some friends and fellow writers like Eric Rucker Eddison (whose novel The Worm Ouroboros also was a favorite of Lewis's) and Dorothy Sayers (the Lord Peter Wimsey mystery series) made guest appearances.
Was there any special motivation for it, and how did it grow to its later glory?
The Inklings liked hanging out together and hoped to encourage each other in their meetings.
As for why Lewis and Tolkien set up the group in the first place - probably for discussion about the works they were writing, and connecting to other writers. Why do you set up a chess club? For practice, fun, and people. It seems like that would be why they would set up a group like that.
This is basically what it says in The Inklings by Humphrey Carpenter:
who edited the university magazine Isis and published a couple of novels while still studying for his degree. There were also a few dons present at the meetings. The club existed so that members could read unpublished compositions aloud, and ask for comments and criticisms. Tangye Lean named it 'The Inklings'.
No record of its proceedings survives, though Tolkien recalled that
Thanks @doppelgreener for all of the transcriptions in chat!