If the speaker of this poem were, indeed, William Carlos Williams, he would have been in his fifties, married, and a father when it was published. So, we'd guess that he would be writing this note to his wife before he went off to his day job as a doctor. Right? Right.
But it's dangerous to assume that the writer of a poem is its speaker, because poets often write about the lives of others. So we'll just have to hazard some guesses about our guy, based on the (meager) evidence we get from the poem itself.
The poem lets us know just a little bit about our guy: he's nice enough to write a note to apologize for eating these plums, but easily tempted enough to eat them. He also puts great value in simple things, like the taste of these plums, and their temperature.
Let's use our imaginations to guess a little more about our speaker. We can picture our speaker waking up on a balmy morning, and heading down to get his normal breakfast of cereal when he sees these plums, and he just can't resist them. We're guessing he's got a steady job—he could be a lawyer, a banker, a dentist—whatever he is, he's got to be there pretty early.
He'll be gone, at work, when his wife wakes up to find her breakfast missing. He feels guilty about eating the plums, and leaving her with none, and figures he'll write a note so that she knows he's sorry. He could also feel a little lonely, waking up in the morning without her, and wants her to know he's thinking about her.
The speaker takes great stock in his own words—he's written this note very carefully, with the fewest words possible. He goes slowly, building up to what he considers the most important: that the plums were delicious.
We like to think that, as the speaker goes about his day, the memory of the taste of these plums will stay with him, giving him a little bit of sweetness and cold even as the day drags on. He hopes that this note, and its description of the plums, will give his wife bit of joy too.