Plot-wise, there's both a lot and a little that goes on here. If we're going by the actual events of the poem, the summary is short and not-so-sweet: young man returns from hunting, and answers a round of questions from his mother.
The real story, however, is the cryptic narrative that Lord Randall offers in response to his mom's queries. He reveals that he went hunting, then had dinner with his special lady friend, who fed him a dish of eels in broth. His dogs, who probably ate some of the same food, died rather horribly. This leads Lord Randall's mother to the terrible conclusion that he has been poisoned, like the dogs.
Lord Randall confirms this, and, stating that he's "sick at heart," also confirms that it was his treacherous "true love" who poisoned him. Yikes. We discover at the end that his refrain, "make my bed soon" refers to the fact that he is "weary" because he's actually going to die soon (in olden days, talk of making beds or going to bed was often a way of referring to death, so the bed in question is either a deathbed or a grave, depending on circumstances).