The first, longer, poem concerns the Norse heroes Sigmund and Sigurd, in stories quite different from the stories in the _Nibelungenlied_ or the _Ring des Nibelungen_. While Odin (pardon; Odín, but I'm going to stop using the accent marks except for first introduction of a name because they're a big pain on this keyboard) does indeed manipulate the events - indeed, in a much more hands-on manner than in the operas - there's no Rhinegold, no building of Valhalla (Valhöll), and while there are many cognate scenes they are, not only in detail but in significance, quite different.
The second takes place after the deaths of Sigurd and Brynhild at the hands of the Gjúkings (well, Brynhild dies by her own hand, but it's the fault of the Gyuking women really). Gudrún, who was briefly married to Sigurd, becomes "wild and witless" for a time, then settles down to weave a tapestry summarizing the first poem. But her brothers and scheming mother determine that she must marry Atli (Attila the Hun) for political reasons. She resists at first but ultimately cannot refuse her mother. And all is well until Atli decides that he wants the Nibelungs' gold (complicated, it was the hoard of Fáfnir - no Fasolt, here, though he *does* murder a brother- and wound up with the Gyukings, who are apparently the Nibelungs also...), and comes up with a scheme to get it by ransoming the brothers. It winds up with the brothers all dead, and Atli murdering Atli's sons, feeding their blood to him, and murdering him in his sleep.
Good clean fun in the nifty Norse manner.
The poems are written in stanzas (staves?) of eight alliterative half-lines each, quite skillfully wrought. There are occasional variations in the stanza form, but most of the poem by far is in this form and it never becomes tedious.
Then there is the (inevitable) apparatus surrounding them. Christopher Tolkien (whom I shall refer to as "Christopher" and his father as "Tollkien" because why not?) provides a lengthy introduction, incorporating Tolkien's various notes on the Eddaic and other Norse poems, as well as the Nibelunginlied - oddly, there are almost no notes or workings surviving regarding the two poems.
In this introduction, as well as the copious footnotes and the first Appendix, Christopher uses available lecture notes and similar to reconstruct Tolkien's views of the origins of and "proper" content of the stories here related. As a result, we do not get Christopher's opinions on the matter; nor do we get Tolkien's opinions on the matter; what we get is Christopher's (well-informed) opinions of Tolkien's opinions on the matter.
...which has, really, been true of almost everything Christopher has edited since his father's death in 1974. I'm just dumb enough not to have realized it till now. Never really thought about it.