How can we be sure that a certain historical claim is true and accurate?

This is a very valid question and the answer, traditionally, is that the historian uses their judgement in combination with logic and collateral information to evaluate claims.

First of all, many historians make no claim as to the authenticity of the information they are reporting, they just report it. For example, Herodotus will just say "so-and-so told me thus and such". Sometimes he will say "I found this unbelievable because..." and then give his reasons.

Many historians do not even do this, they just accept what they read and repeat it uncritically. For this reason you can find the same mistake being repeated over and over in many books. For example, the main source of information on the American Revolution is the book Storia della guerra dell' Independenza d'America (1809) by the Italian Carlo Botta. This book was translated into English and widely used by American historians to "document" the American revolution, even though Botta did not speak good English and had never even been to America. Consequently, American history books are loaded with incorrect or garbled information that Botta either made up or exaggerated.

To answer your question more specifically, a historian can infer what happened by using logic and collateral information. For example, you will see the claim sometimes that the Soviets colonized Prussia with Russians after they conquered it in World War II, but if you actually go to Kaliningrad, formerly known as Konigsberg, you will find many Poles there, and they are in fact by far the dominant group. From this you could infer that a systematic effort was made to introduce Poles into conquered Prussia, even if you did not know this from documentary evidence. You can use collateral evidence to infer what must have happened, even if it is undocumented or documented incorrectly.

While historical veracity may have, at one time, depended on a multitude of written historical texts as well as the "authority" of the historian, history is, thankfully, increasingly multidisciplinary:

Biologists use DNA to ground truth population dynamics in ways historians can only dream of. For example, an entire Biblical story line can be discarded as nonsense by DNA analysis of Middle Eastern peoples.

Palynologists studying lake sediments can determine a regional climate, local flora, and even determine the carrying capacity of a region for feeding it's people. Was it possible for this bronze age city to really have hundreds of thousands of citizens at the same time, despite what historical texts say?

Microscopists use sophisticated imaging techniques to ferret out invisible details from artifacts. Steadily degrading mint marks in Roman coins indicate inflation and overproduction of coins to combat it. Hard proof that citizens were being ridden hard by their government and were very unhappy, despite whatever propaganda Cato had written to the contrary.

I love the story of how, practically overnight, nuclear chemists used carbon dating to overturn a thousand years of "expert" historical analysis that said all European culture was derived from the Greeks.

Some history may be forever unknowable, but increasingly, other disciplines are providing extremely useful tools that can verify what historians think they know and discard what is bias, propaganda, lies, fantasy, or outright nonsense.

They have to be careful, and even so there are controversies and mistakes.

• What is known about the author, and did he or she have an agenda? Consider the Donation of Constantine.
• Do any artifacts support the story? Schliemann's discovery of Troy is an example.
• Are there any other sources? Would one expect other sources if the story was real? Extraordinary claims require extraordinary evidence.