Before directly answering this question, it is important to understand the position of the men as Beowulf's thanes. Throughout the poem, the bond between thane and treasure-giver, or king, is manifested to be a strong and inviolable one. The king gives his thanes shelter and food and riches, and they in return swear loyalty and obedience. When the king is attacked, the thanes must defend him; and when the thanes perform such deeds of valor, they are rewarded with treasure.
Beowulf's men, however, hide when it comes to fighting the dragon. It is true that Beowulf tells them not to help him, but this is confusing; they must keep their oaths of obedience but not of loyalty. They must obey but not rally round the king, yet the one necessarily includes the other. The choice to accept this contradiction seems to be motivated by their unmistakable fear. They prefer to leave the monster to Beowulf and hide behind the command.
However, Wiglaf, one of the thanes and a relative of Beowulf, cannot stand this. He speaks chidingly to the others:
I remember the time that we took mead together,
when we made promises to our prince
that we would pay him back for this battle-gear,
these helmets and hard swords, if such a need
as this ever befell him." (Liuzza, lines 2633-38)
In other words, we pledged to defend our king. He gave us the weapons in order to defend him; now we have to fulfill that debt by so defending. Wiglaf adds, "For this he chose us from the army / for this adventure" (2638-39); that is, he selected us to accompany him precisely because of our pledges of loyalty. After urging the men to fight, he tells them, "It seems wrong to me that we should bear shields / back to our land, unless we first might / finish off this foe, defend the life / of the prince of the Weders" (2653-56). Can the battle-gear be carried home honorably when we have refused to use it as we promised?
Nevertheless, after thus pointing out that their duty now lay in being loyal rather than in obeying, Wiglaf was the only man to join the fight. The men hid, in fact, until not only the dragon, but also Beowulf was dead. This fact is significant, for their cowardice in approaching a living, although dying, King shows that their presenting of themselves would be evidence and a reminder of their desertion and base ingratitude.
What does their behavior mean for themselves and for the kingdom? In the first place, they destroyed the bond which formed the fundamental basis of their warrior society. Instead of remaining bound in a tightly unified social structure, they preferred to tear it apart by seeking their individual wellbeing over the common good--the preservation of their leader and the upholding of the trust. A society which disregards the whole for the sake of the component parts is fundamentally fragile.
Trust has been destroyed. The oath has been violated. The debt of loyalty has been disregarded for selfish reasons. With such a precedent, the kingdom begins weakly, not just because Beowulf is dead, but rather because the bases of unity and of order have lost their sacred character. The poet himself foretells violence, conflict, and destruction.