I'm sorry to tell you, but you've asked an enormous question that historians, philosophers, scientists, and others have struggled with for, basically forever. Herodotus and Thucydides were aware of these issues, and both did what they could to make their accounts of history "accurate," while both acknowledged their work was far from perfect.

Herodotus, of course, should probably be criticised more than Thucydides, since Herodotus apparently didn't bat an eye when told that Giant Ants (likely a lost in translation issue over large marmots) were digging up gold in Eastern provinces of Persia.

Marwick and White debated even the means of historical communication, so to answer your question: Only with great difficulty.

It's very, very difficult.

I study a period that had very few texts, so I use a lot of archaeological evidence. Some archaeologists think it's possible to recover what people in the past were thinking from their archaeological remains (ie, Cognitive Archaeology), but most agree that it's impossible. You can, however, often see what people cared about by looking at what they chose to do. We don't necessarily know what Egyptians were thinking when they mummified cats, but we know that they thought dead cats were important because they went to the effort of preserving them. So you can learn a lot about what mattered to societies by studying what they did, even if you don't know exactly why it mattered to them so much. That's how I try to determine which facts were relevant within the past society I study (6th century England).

Now, when you have documents, you can do a lot more because people often write about what they think matters. Each kind of document has its own limitations and difficulties, though. For example, ancient and medieval history writing was usually trying to tell a moral lesson, so you can't study the 'facts' it reports without considering them in the context of the moral the author was trying to draw (ie, did it really happen that way, or was the author manipulating the 'facts' to make his point? Did everyone care about the stuff the author cared about?). Law codes: do they tell us how people actually acted, or how rulers wanted them to act? Census data: did they count every person in a region, or only the important people? Etc. And then there's the question of power: how do you recover the voices of the 99% when most of the texts were written by the 1% at the top? Do we ever really know what the 'average' person cared about in the past, when they left so little evidence? Every kind of document has its own difficulties, and historians have a lot of tools and tricks to read between the lines, against the grain, etc. Learning these tools is one of the reasons why history PhDs spend 8 or 9 years in grad school to get their degrees.

Historians rely a lot on consensus. Every historian works on a small part of the puzzle, and compares their part to the work everyone else is doing (reading this material is another reason history degrees take so long to complete). As historians study and write and share, we can gradually assemble more information about the subject we're studying. Slowly, the big picture starts to emerge. Recently, there's been a strong push to be more interdisciplinary in our approaches, so historians are looking not only at texts, but also at art, literature, archaeology, etc, to get as much information as possible. But we don't ever produce a perfect picture, just interpretations of the available evidence.

The best history doesn't just tell you 'facts'. It also tells you how the author studied the sources to gather this information, so you can evaluate for yourself if s/he missed something or asked the right questions. Most of this information is usually in the footnotes.