By using multiple sources, including non-litetary sources, for one. By using sources critically, for another. Today especially, there are far more ways to gather historical evidence than simply by relying on one culture's literary record. Archeological Science has advanced incredibly over the past century, and the same can be said for atomic science (which we have to thank for radio carbon dating for example) and climate science, while new, is also providing us with critical new evidence with which to form our picture of the past. We can examine material culture, burial grounds-both in terms of things like grave-goods and ceremonial characteristics, and forensic examination of the body-, structural foundation remains, etc. And we can look at surviving literary sources from adjacent literate cultures. We can compare a given literary source's evidence against all of these non literary sources of evidence in order to evaluate the reliability of the author and (when we are lucky) the veracity of specific claims of the narrative. And, of course, using a source critically means learning what you can about the author in order to form a sense of bias, authenticity, etc.

Of course, none of this is simple. Doing the work involved is the job of historians. It really is more like a science than a humanity when you approach it correctly. You observe a phenomenon, form a research question to better understand the phenomenon, look for data to inform your research question, and seek to interpret that data in order to answer your research question. And, like hard sciences, hypothesis take time and effort and evidence before they are accepted as viable theories, and often necessarily evolve as the historian conducts their research.

The historical method is based on the critical reading of historical sources. When studying sources, we look for bias and corroboration. Bias is when the author of a given sources has interests in portraying events a certain way. Nearly all sources are biased in one way or another, but we cannot discard the claims of a source simply because of this. However, when a source contradicts its own bias, it becomes very interesting. If, say, Emperor Augustus paid for an engraved wall, which praised his greatness and magnanimity, but also conceded that he lost a battle, we can be quite sure that he actually lost the battle. However, we would need someone to corroborate on his greatness and magnanimity.

This is where corroboration comes in. If we also had another source, say, a private letter from a soldier to his brother, which claimed that Augustus always paid the wages of his soldiers on time, and gave great bonuses whenever they won a battle, then we can also trust Augustus claim of magnanimity. If, however, the soldier claimed that Augustus was a miserly prick, who never paid on time, then we would have to weigh the two sources against each other. In this instance, Augustus' wall, essentially a piece of propaganda, would probably be much more likely to lie than an informal, private letter from a soldier to his brother. Simply put, the latter would not have any reason to lie (that we know of), while Augustus would have an interest in portraying himself in as positive a light as possible.

However, there are a lot of unknowns in this example. What if the soldier had a personal beef with Augustus, and wanted to slander him? What if the soldier hated his brother, and wanted him to miss out on a shitload of money out of spite? There are certainly a lot of unknowns whenever you work with a limited amount of sources, and you become less precise as a historian when you have fewer sources about a given subject. I hope you can use this answer :)

As others have commented, finding multiple sources from different perspectives is a major way to 'vet' any one particular source. One of the interesting things I found while researching Gen. Thomas Kane was a biography written by his granddaughter. His granddaughter insisted that General Kane (who is most known in the West for being an ally to the Mormons) didn't know the Mormons were polygamous until quite some time after helping them form their own battalion in the US Military (in the late 1840's). It is an interesting claim, because at the time General Kane was meeting with Brigham Young and other leaders for extended periods. So, if true, it was somewhat of a stretch if General Kane was collaborating so closely with the Mormon leadership to help them move west to Utah. It's been a bit too long since I did the original research footwork, and whether or not General Kane knew about polygamy in the 1840's wasn't really my main research question, so I don't have a complete answer to 'true or not.' However, as with other posts, here you would look for other sources that talk about General Kane's relationship with the Mormons. The papers of one or more Mormon leaders could be handy, if they got detailed enough to mention if they introduced Kane to their (multiple) wives, as would more general research about the national talk of Mormons at the time. If newspapers were doing stories or cartoons that mentioned the plural wives aspect of Mormonism around this time, Sybil's claim that Thomas Kane didn't know would be suspect too. Finally, Kane helped lobby for the creation of the Mormon Battalion back in Washington DC, so if anybody attending those meetings kept notes, those could help see if polygamy came up there as well.

I'm on my phone, so I won't pull it up, but I know "Liberty to the Downtrodden" is a good, historical biography for General Kane that may have actually answered this tidbit. But even if Sybil's claim incorrect (by accident or on purpose, she was his granddaughter after all), it does connect to the broader conversation about Mormonism in America (or the "Mormon Question" as it has been called). So even when a source isn't historically accurate, they can be very valuable in other ways!

Edited some grammar mistakes.