One of the first aspects of an epic poem one examines to learn about the society the poem derives from is the characterization of the hero: specifically, what makes the hero a hero.
In the case of Beowulf, scholars assume that the character traits that make Beowulf a hero were valuable to Anglo-Saxon society. Beowulf is brave, honorable, respectful of his father and ancestors, a great warrior, and believes his good deeds and great victories are his means of immortality (he will be remembered). And though he appears to us as arrogant, he does give great respect to others and acknowledges the role of fate. Scholars assume, then, that these traits were important to the Anglo-Saxons.
More important, however, is what the poem reveals about the unstable life of the Anglo-Saxons. There was no central government, police force, legal system, etc. in Anglo-Saxon England. Kings weren't kings in the way we think of them today. There were no knights as we think of them or as they appear in King Arthur myths (not until the Middle Ages), and chivalry was centuries away from being thought of. Knights and chivalry are medieval ideas, not Anglo-Saxon ideas. And feudalism came to England via the Normans in 1066. Anglo-Saxson society was unstable.
Kings were kings over a bit of land and some people, but over a hill or two was another king who ruled his bit of land, etc. All it took for a person to be displaced was for one mead hall to attack and take over another mead hall. In other Anglo-Saxon works like "The Seafarer" and "The Wanderer," this seems to be the central problem in Anglo-Saxon life.
And, essentially, this is what happens in Beowulf. What is Grendel? He is every Anglo-Saxon's nightmare. He is the mead hall wrecker who displaces mead hall residents, and turns them into exiles and wanderers. He is the worst thing that could happen, and often did happen, to Anglo-Saxons.
The Beowulf poet uses a monster to demonstrate the unstable nature of Anglo-Saxon existence. Grendel may be a monster, but he does only what Anglo-Saxon kings often did to other Anglo-Saxon kings. This is what is revealed in the epic poem. The monster may be a literary creation, but the destruction of a mead hall and the displacement of the people who depended on the hall and the king for protection is not.
There are a couple of features of Beowulf's characterization that reflect what was important in Anglo-Saxon society. For example, Beowulf comes across as rather brash and egotistical, but Anglo-Saxon society valued the daring and confident young man who was proud of his actions and could use his past endevours to motivate himself towards future success. Our modern sensibility is towards modesty, but not then! They told and retold the great stories to remind themselves and others of their greatness.
Anglo-Saxon society believed in a concept revenge that resulted in a revenge cycle. If someone killed one of your men, you were obligated to avenge his death, but then that group was obligated to avenge that death -- and so on and so on. this is evidenced in Beowulf's killing of Grendel, but Grendel's mother seeking retribution. The cycle only ends because there is no one left to carry out the revenge.
Anglo-Saxons also believed in the concept of fate, called wyrd. There are numerous references to wyrd in the text, so while the characters take actions to defeat their enemies, they are also aware that fate has a hand in the situations and outcomes of man.
One of the most important ways that Beowulf reveals the values of the Anglo Saxon society is in its chivalry code, and the code of the knighthood.
Back in Anglo Saxon times, it was common of the people to desire a King that was chivalrous, powerful, and daring. This was the early feudal society that built up what later would become the middle ages. From this paradigm, the people would, in turn, vow their loyalty to the king by becoming his knights and defenders. As this progresses, a new hierarchy forms, and the people of a town have a complete system of defense all based on the honor and loyalty that they feel for the King.
As an Anglo Saxon poem based on oral tradition and history, Beowild woulod comprise all the Anglo Saxon elements that would make it reveal society as it was.