What lessons would the people in the time of the poet have learned from the story?
It's probably important to note that at the time of the poet there weren't have been readers, just listeners. Most estimated timelines suggest that this story wasn't written down until 200 years after its first telling.
That aside, other than the great action of the story, Anglo-Saxon listeners would have appreciated the inclusion of their societal values into the story. Beowulf and his followers all exhibit them at some point: loyalty, generosity, bravery, boasting, and obligation. There are numerous lines throughout the epic that show these traits.
Through the inclusion of these traits, the A-S audience would have learned that their heros were still required to show the same characteristics that they were. It would be easier to show generosity to your fellow tribesman if you know that people like Beowulf were doing it also.
The lessons included loyalty to leaders and bravery in battle. Beowulf is a strong, fierce fighter who fights alone. He has the bravery and the strength to do this. He wins. The lesson is - be strong and be brave. It is only at the end that Wiglaf helps him in battle. It is in this battle that Beowulf dies. But thats ok, because Wiglaf is there to take the mantle of leadership. Beowulf died a hero because he intended to fight alone and because he still showed bravery. The qualities of a hero were very important in the Dark Ages, as leaders helped to bring stability and safety to the people.