How is progress in history made?

Progress in history can only be made in retrospect. Hindsight is innately associated with history, and with this you can better interpret the past. As time moves forward, our understanding of the past as a whole becomes much greater, even if our understanding of one particular person/place/thing remains constant. e.g. better understanding of pathology might give insights into possible better understanding of Town A's past disease outbreaks, which in turn could have led to Town B's logic in doing XYZ.

The more time that passes the greater our scope on the past becomes, which often leads to better understanding of what was taking place in the grand scheme of things. Inversely, as our scope of the past gets wider, our methods of fact-checking get narrower, our information becomes more stale, and eventually we begin relying more on interpretations from others in place of the exact truth. So, as the widely stretched or vague interpretations that fill the gaps of history are made clearer through technological advances, so to is our 'progress' towards a more thorough understanding.

A positive attitude complements the study of history. Without contradicting Mark Olson's excellent answer, I think Charles-Edwards was demonstrating an appropriate optimism about his fellow historians' promise. To claim that nothing more will be known is not only likely to be wrong, it's a downer.

Don't forget, the study of history itself can and has advanced.

Before we had writing, we relied on oral tradition to pass down historical knowledge. This was prone to degradation - ala the telephone game - and besides, the goal was not necessarily to preserve objective truth, but usually to tell a good story. Real figures get mythologised, aggrandised and combined, animals gain extra limbs, and stories had to have heroes, villains, lessons to tell and so on.

Herodotus, the "father of history", was a key figure in changing that, and so people became interested in knowing history for its own sake, started to question those oral myths, differentiate sources by reliability, and look for hard evidence.

But pre-Enlightenment histories were often heavily biased. Herodotus attributed supernatural causes to many events. Official histories in Imperial China often cast the previous dynasty - especially the latter emperors - to be morally corrupt, justifying their rule using the Mandate of Heaven narrative. Historians such as Voltaire brought these issues to light, of the political bias - and sometimes outright fabrication of evidence - prevalent in traditional records, as well as demonstrating new ways of looking at history, apart from biographies of important men and chronicles of polities.

In more recent times there has been an explosion of different schools of historiography, and different ways of studying history. Marxist historiography recasts history through a Marxist lens, and how history is affected by conflicts between social classes. Feminist history focuses on the effect of patriarchal narratives, forcing us to take a second look at women who were perhaps unfairly judged - like Cixi - or ignored entirely. Each fresh perspective offers opportunities for discovering new things hidden in known sources.

It is therefore easy to conclude that the study of history will continue advancing; it would be hubris to assume we are at the end of its development. For a fictional example of where we may go, consider Asimov's psychohistory, which applies statistics to history in order to extrapolate future events. But it might also be used to interpolate known, past events, and discover anomalies that change our knowledge of history.

When history isn't clear, it can be advanced in the same way science is advanced when conclusive evidence through experimentation cannot be obtained.

Historians formulate theories about the subject. Then they try to support their theories by finding corroborating evidence. Given theory A then X, Y and Z must have existed. Then Historians go out into the field and try to find archeological evidence for X, Y and Z. If enough corroborating evidence can be found and a convincing case assembled, it is presented to historical committees made up of respected experts of the period and they judge the voracity of the case. Sometimes the originators of a theory are not the ones who transform the theory into accepted history. Sometimes that is done decades, even centuries, later with the help of facts or science unavailable to the original historian.

As such the search for Arthur goes beyond just trying to uncover new historical texts. Every time archeologists discover previously unknown fortifications, and evidence of advanced commerce at sites associated with Arthur, knowledge of the mirky period in Britain's history progresses, and light is shed on Arthur. The ruins at Tintagel and Chester are such examples.

National Geographic (August 2016)

A recent discovery in southwest England is making headlines for its association with King Arthur, but archaeologists are hailing it as an incredibly important find regardless of any connection with Britain's greatest legendary ruler.

Excavations at Tintagel, a rocky promontory on the coast of Cornwall, have revealed evidence of massive stone fortifications and luxury goods imported from as far away as modern-day Turkey, all dating to a poorly understood period in British history that began with the collapse of Roman rule on the island around 400 A.D.
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According to 12th century historian Geoffrey (of Monmouth's Historia Regum Britannie- "History of the Kings of Britain"), Arthur was conceived at Tintagel after his father, Uther Pendragon, disguised himself (with the help of Merlin) as a local ruler and slept with the ruler's wife.

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The Telegraph July 11, 2010
Historians locate King Arthur's Round Table Historians claim to have finally located the site of King Arthur’s Round Table – and believe it could have seated 1,000 people.

Researchers exploring the legend of Britain’s most famous Knight believe his stronghold of Camelot was built on the site of a recently discovered Roman amphitheatre in Chester. Legend has it that his Knights would gather before battle at a round table where they would receive instructions from their King. But rather than it being a piece of furniture, historians believe it would have been a vast wood and stone structure which would have allowed more than 1,000 of his followers to gather. Historians believe regional noblemen would have sat in the front row of a circular meeting place, with lower ranked subjects on stone benches grouped around the outside. They claim rather than Camelot being a purpose built castle, it would have been housed in a structure already built and left over by the Romans.

Some of the theories compiled around Arthur are truly fantastic. Take the ongoing quest for the Holy Grail which legend has it was found by the Arthurian knight Sir Galahad, and according to one amateur historian recently rediscovered, in a loft in Rugby.

the Independent August 10, 1995

One man's quest for the Holy Grail ends in a loft in Rugby

An amateur historian claims to have tracked down the sacred relic after years of research.

The extraordinary find came to light after seven years of investigations by a Coventry-based historical researcher, Dr Graham Phillips, who tracked the potential "relic" down to a house in Rugby, Warwickshire, where its owner kept it in a box in the loft.

Dr Phillips has unearthed a substantial body of evidence linking the find - a small green onyx cup of possible Roman date - to the Grail legend.

In the medieval Arthurian romances, the Holy Grail was the cup used by the man who buried Jesus - Joseph of Arimathea - to collect Christ's blood. However, the origin of the medieval legend appears to have been a real historical grail "found' in the 4th century AD by the newly-christianized Roman imperial authorities who turned it into a sacred relic.

According to the 5th century Greek historian, Olympiodorus, this original Holy Grail had been used by Mary Magdalene to collect Christ's blood and was found by the Christian Roman empress, Helena, inside Christ's Holy Sepulcher.

Saint Helena, or Constantine the Great's mother, who also found according to Roman and early Catholic sources the cross which Jesus was crucified upon. Constantine the Great ordered the Church of the Holy Sepulcher, built upon the site where his mother found said cross.

Wikipedia
Constantine appointed his mother Helena as Augusta Imperatrix, and gave her unlimited access to the imperial treasury in order to locate the relics of Judeo-Christian tradition. In 326-28 Helena undertook a trip to the Holy Places in Palestine. According to Eusebius of Caesarea (260/265 – 339/340), who records the details of her pilgrimage to Palestine and other eastern provinces, she was responsible for the construction or beautification of two churches, the Church of the Nativity, Bethlehem, and the Church of Eleona on the Mount of Olives, sites of Christ's birth and ascension, respectively. Local founding legend attributes to Helena's orders the construction of a church in Egypt to identify the Burning Bush of Sinai. The chapel at Saint Catherine's Monastery—often referred to as the Chapel of Saint Helen—is dated to the year AD 330.