Owen Barfield deserves his description as 'the fourth Inkling' - along with CS Lewis, JRR Tolkien and Charles Williams - because, although he attended few meetings, his influence is undeniable (especially on Lewis, but also on Tolkien) and his core philosophical concern was exactly that shared by the other three Inklings.
Indeed, Barfield stated this primary concern more explicitly and over a longer period than any other Inkling (because he started publishing so young and lived an active life up to the age of ninety-nine).
This theme is the Truth of Imagination. Barfield's life-long concern was to understand how the Imagination is a source of Truth, a source of knowledge, a way of accessing reality.
This was also Tolkien's concern, most evident in the concept of Subcreation described in his essay On Fairy Stories - and in his many reflections on myth and history.
It was Lewis's concern in his Platonism - where the Imagination was seen as a mortal and earth-bound way of understanding the primary eternal forms of Heaven - this crops up all through Lewis's ouvre - for instance at the end of The Last Battle and his book on the Medieval world view - The Discarded Image.
And Charles Williams many considerations of Romantic or Positive Theology (via positiva) - his multi-form efforts to show that the poetic imagination could be a path towards salvation and theosis; and his best and most explicitly Platonic novel The Place Of The Lion - in which the imagination opens-up a (dangerous, indeed deadly) channel for the eternal forms to invade this world. Furthermore, in his actual life, and to a high degree, Williams lived by the truth and reality of imagination.
Of course, this concern with the Truth of Imagination was mostly a matter of the confluence of spontaneous personal interests rather than of direct personal 'influence' of one Inking upon another - especially in the case of Charles Williams whose ideas were fully expressed before he even heard of the Inklings (in 1936), and before he actually attended meetings regularly (from 1939-45).
By contrast with Williams, Barfield had done most of his thinking and formulating back in the 1920s, before The Inklings, around the time of his Great War with Lewis (a sustained epistolatory debate from 1925 to Lewis's conversion circa 1930); and when the friendship was forming between Lewis and Tolkien. Barfield's early BLitt thesis, and his first two books (Poetic Diction and History In English Words) also (by his own account) influenced and changed Tolkien at this time - both as a professional philologist and in his imaginative writing.
Once we are sufficiently clear about the nature of The Inklings primary concern, the importance of Owen Barfield becomes obvious.