The series is based on the first two Barchester novels, The Warden (1855) and Barchester Towers (1857) which, you may or may not know, focus on an otherworldly clergyman, Septimus Harding, who is caught up in a newspaper-driven controversy over his occupation of a well-paid post of nominal responsibility, warden of the almshouse Hiram's Hospital. At first, Harding accepts the patronage of those helping him to stay in the post (and to reinstall him when it becomes vacant), but later he comes to question the ethics of the situation. The Victorian Web has a good appreciation and summary by Annalise K Walker of the full series: On Trollope's Barchester Series.
I had somewhat wondered where, if anywhere, Trollope's fictional "Barsetshire" is set. The Warden begins with a basic description ...
the cathedral town of ––––; let us call it Barchester. Were we to name Wells or Salisbury, Exeter, Hereford, or Gloucester, it might be presumed that something personal was intended; and as this tale will refer mainly to the cathedral dignitaries of the town in question, we are anxious that no personality may be suspected. Let us presume that Barchester is a quiet town in the West of England ...... but unlike Thomas Hardy, who interacted with readers and critics in mapping "Wessex" highly consistently on to real locations, Trollope didn't create a detailed geography at first, but sketched it out later as he wrote the 1860 Framley Parsonage. The paper Mapmaking in Barsetshire (Lance O. Tingay, The Trollopian, Vol. 3, No. 1 (Jun., 1948), pp. 19-32) mentions how this didn't come to light until 1927. Prior to this, enthusiasts had produced maps: the USA Trollope Society website has that painted by George F Muendel for Spencer Van Bokkelen Nichols (here), and another was produced by Monsignor Ronald Knox, who later wrote the 1935 novel Barsetshire Pilgrimage
Knox, initially working in 1922, showed that the books were inconsistent with Trollope's sketch, and even with each other. As Gillian Hill's 1978 Cartographical Curiosities (British Museum Publications) describes:
Trollope's Barsetshire, on the other hand, suffers from an excess of local topographical information. The general picture of the countryside is fairly clear, but contradictions abound in the detail. Ronald Knox brought these to light when he attempted to map the county in 1922, more than fifty years after the novels were written. At the beginning of the first of the Barsetshire novels, The Warden, the cathedral is at the west end of Barchester, but in the later novels it has moved eastwards, and stands by the London Road. Similarly, the distance between Barchester and Plumstead shrinks from nine miles in Barchester Towers to less than five in The Last Chronicle of Barset. The railway, too, has its idiosyncrasies: it passes within a mile and half of Courcy, which is a considerable way from its route.The Tingay paper mentions many more such cartographic problems. Later maps include Morris Weightman's map (here) for Angela Thirkell's 1930s Barsetshire novels. But inconsistencies apart, details such as train routes - eastward to Paddington, north-west to Bristol, and south-west to Exeter - plausibly put Barsetshire somewhere in the general area of the Wiltshire / Dorset boundary.
Trollope's own accounts of the inspiration also give geographical clues. The standard story in his Autobiography, as told in Trollope and the Close (Salisbury Cathedral Close Preservation Society, Annual Lecture - October 2006 - by Pamela Neville-Sington) was that it happened in Salisbury:
I had stood for an hour on the little bridge in Salisbury, and had made out to my own satisfaction the spot on which Hiram's hospital should stand.But he was also inconsistent on the matter, and in the last weeks of his life he told the historian EA Freeman that Barchester was based on Winchester (where he had attended school) and that Hiram's Hospital was based on St Cross. This is consistent with the chronology of a mid-Victorian scandal concerning Francis North (nepotistically appointed as Master of St Cross by his father, the Bishop of Winchester) misappropriating funds. However, it's by no means the only contemporary scandal. There's a very good analysis of the whole background in the notes to the Broadview Press edition of The Warden (Anthony Trollope, ed. Geoffrey Hardy, Broadview Press, 2011, ISBN 1551111381) - see, for example, "The origins of Barchester" (from page 9), "Trollope's comments on the genesis of The Warden" (from page 237), and "The major scandals alluded to in The Warden" (from page 247).
I mention all this background not out of viewing it as some definitive "meaning" to the novels, but out of the way it gives a handle of the social history of the period. Victorian England was a time of vast social change, and the conflict between the reforming and evangelical "low church" and the "high church" (the Anglican church run by a long-entrenched moneyed elite) was a major topic of debate, especially in the provinces.
In Exeter, for example, Henry Phillpotts (Bishop of Exeter, 1830 to 1869) was a classically controversial high church bishop who commanded a vast income by astutely gaming the preferment system. At a time when a grass-roots clergyman might earn £40 a year, he swung a non-residential canonry at Durham worth around £4000 a year in addition to the £3000 he earned as Bishop of Exeter, an appointment highly unpopular locally); this double income funded the building of his residence at Torquay (now the Palace Hotel). Niebuhr and Wilkinson's 1956 The Ministry in Historical Perspectives described him as one of the last of "a clerical aristocracy, who, whatever their origin, expected to live on a scale comparable to that of the nobility". He was satirized in the pseudonymous 1832 pamphlet Bishop Toby's Pilgrimage; Or, The Method of Procuring a Mitre., and gets a brief allusion in The Warden as Henry, one of the three sons of Dr Grantly.
Addendum. Small world. Angela at Literary Places just sent me the detail, from WHK Wright's West-Country Poets (page 167) that Bishop Henry was the great uncle of the writer Eden Phillpotts. I'd wondered if there was a connection, but was busy pursuing the Barchester thread.