Being remembered after death—it's just what makes the speaker of "Remember" tick. She keeps telling her beloved to remember her, because, well, she's obsessed with death. Okay maybe obsessed is putting it a little strongly, but she's clearly somebody who is really thinking a lot about death.
And why might that be? Well, for one thing, this poem was written in the middle of the nineteenth century. People died much more suddenly, and much more frequently, than they do nowadays. That's one reason. Another reason may be because this speaker has a few, well… let's just call them hangups. She's not quite goth or EMO or anything like that, but she's definitely thinking about death a heck of a lot, but this isn't necessarily a bad thing. The speaker is at least realistic.
And not only that, she's definitely a very caring person. Sure, the whole "remember me remember me remember me" business might seem a tad selfish, but by the end of the poem the speaker shows us how selfless she actually is. In the poem's final lines, she essentially says "On second thought, it's actually better if you forget about me, because remembering me will only cause you sadness." Wow, talk about the ultimate gesture of selflessness. She would rather the man she loves be happy than remember her. She just can't bear the thought of him being unhappy.
Now we're sure you're wondering if the poem is spoken by a "real" person or not. Technically no, but technically… yes. Let's explain. In poetry, it's never a good idea to confuse the speaker with the poet, even if the poem is written from a first-person perspective ("I," "me"). Here, though, that rule may need a bit of bending. In many ways the speaker of this poem is the young Christina Rossetti, at least a little bit. If you've read our "In a Nutshell" or "Calling Card" sections, you know Rossetti thought about death a lot, and that she was well aware of how short life really can be. Moreover, she suffered a mini nervous breakdown sometime in the 1840s, which likely contributed to some of her, ahem, compulsions.