So that her highborn kinsman came And bore her away from me, (lines 17-18)
The speaker never quite comes out and says he doesn't like Annabel's family, but you can almost hear how crabby he is about this "highborn kinsman." We think there's a bit of a "kid from the wrong side of the tracks" theme here. It's just a hint, but it adds some extra tension and texture to the plot.
To shut her up in a sepulchre In this kingdom by the sea. (lines 19-20)
Death is the thing that first comes between the speaker and his beloved. But then the family comes in and seals the deal, taking her away for good and literally shutting her away from him in a tomb. We almost get the feeling that he thinks the family, and not death, are the bad guys here. In any case the speaker has a pretty paranoid, "us against the world" approach to this whole situation.
But our love it was stronger by far than the love Of those who were older than we--(lines 27-28)
This looks to us like another swipe at those "highborn kinsmen" that he's obviously not so crazy about. Instead of letting himself get pushed around by these older people, he insists that he has something they don't – the strongest love in the world. It's a feeling a lot of us can probably relate to. It's annoying to be told that you must not know what you are talking about because you are young.