It's not like Beowulf is an evil character who doesn't care about the well-being of others, he really does and he doesn't fight monsters just for his benefit. Beowulf is a leader and he is responsible for a large number of people. Therefore, since he is part of a group and groups help each other, it was Beowulf's responsibility to take care of the group. Even though Beowulf is old and his skills are deteriorating by the time he has to fight the dragon, he still feels that he can defeat the monster in order to save and help his people. This is proven by a few lines from the epic which state: "And Beowulf uttered his last boast: 'I never knew fear, when young I fought in endless battles.' I am old now, but I will fight again, and always seek glory, if the dragon hidden in his tower dares to face me. (623-628) Just like he did with the other two monsters, since many people were being killed by them and he helped them even though they weren't even his people. Clearly, Beowulf didn't just care about himself, he cared about others and his people. This is why he continued to fight the three monsters even though his life was at stake.