Why does Hrothgar's "Coast-Watcher" ask who Beowulf and his men are before he is willing to let them land their ship?
During this era (550-900 CE), any Viking ships coming unannounced and unexpected into someone's territory could either be there for good or evil purposes, and it was much easier to fight enemies before they came ashore than on land. Every king who had coastline or rivers that connected to the ocean usually had designated troops to guard the sea approaches to his kingdom. Hrothgar's "coast guard" is simply trying to ascertain whether Beowulf and his men come in peace or anger.
Beowulf's answer is designed to put him at rest:
We are men of the Geatish people/And Hygelac's hearth- companions. My father was well-known to folk, a noble chieftan by the name of Ecgtheow.
Because one's heritage and ancestors--particularly one's king and father--were key to one's identity as an honorable or dishonorable man--it was very important to let a challenger know he had nothing to worry about.
In addition, Beowulf further allays the coast-watcher's concerns by telling him that the Geats have heard of a monster troubling Hrothgar, and they have come to help defeat "some kind of ravager" (Grendel). Relieved that he's not dealing with marauders from Denmark, the coast-watcher not only offers to guide Beowulf and his men to Hrothgar but also makes sure troops are on hand to take care of Beowulf's "wave-floater," the ship.