Literally, there's no room left in the boat after the paddy is loaded, so the paddy cutter can't get on board. As for what that means metaphorically, there's no one answer. Here are some possibilities.
A representation of how a poet's works outlast the poet. The first stanza includes the phrase "The sheaves lie gathered." Sheaves refers of course to the paddy, but we also speak of sheaves of paper. The paddy harvest could represent Tagore's writing. He doesn't know how his works will spread in the world; he doesn't even know where they'll end up:
Oh to what foreign land do you sail?
Come to the bank and moor your boat for a while.
Go where you want to, give where you care to,
But come to the bank a moment, show your smile -
Take away my golden paddy when you sail.
He has sent his poems on their way. He knows his works will make a difference in the world—the goldenness of the paddy has been transferred to the boat itself, and it's now a golden boat. But the ship has sailed, as it were, and he has no control over how his work will be received. They are out of his hands, and he's left behind helpless and alone.
A poetic depiction of one of the best-known tenets of Hindu philosophy. One of the most famous passages from the Bhagavad Geeta says:
कर्मण्येवाधिकारस्ते मा फलेषु कदाचन।
मा कर्मफलहेतुर्भूर्मा ते सङ्गोऽस्त्वकर्मणि॥
karmaNyevaadhikaaraste ma phaleShu kadaachana
maa karmaphalaheturbhuurma te saÑgo.ostvakarmaNi
You have the right only to labor, not ever to the fruits thereof; don't let your motive be the fruits of your labour, nor allow yourself to be idle.
The idea is that our actions are within our control, but the results aren't. To take one example: all I can do is vote for my preferred candidate; I can't control the outcome of the election. If I vote with the idea that my preferred candidate must win, I will be disappointed. But I shouldn't just say, "what difference does it make whether I vote or not?" and stay home either.
The paddy farmer's story exemplifies this. It's his job to harvest the paddy and send it along. He has no control over what happens to the paddy, but he still has to do his job even though he can't sail away with the harvest and enjoy the results. This interpretation isn't very different from the first; it's a more generalized way of saying the same thing.
Romantic love. The paddy farmer has to do whatever he can to win the love of the boat-woman, whom he knows and wants to woo, but while he gives her his all, she has no room in her boat/heart for him. This would be a more convincing interpretation if the Bengali original actually had the same pronouns as William Radice's translation. Radice is hands down the best translator Tagore has ever had. But Bengali pronouns lack gender, and a Bengali reader would find Radice's choice to translate চিনি উহারে / chini uhaare as "I know her" rather than "I know him" a bit unusual. Culturally, the Gangetic delta has boatmen, not boat-women. Of course, if Radice had used a male pronoun, this lovelorn interpretation probably wouldn't occur to most readers, because heteronormativity.
A symbolic portrait of death. The paddy, which the narrator has tended, is now ready to be gathered. The paddy is his soul, and its being harvested means that his life has come to an end. The boatman (or woman) is a Charon-like figure. The narrator's being unable to stay with the paddy means he is separated from his soul, i.e., he is dead. He doesn't know what is in store after death, so doesn't know to what foreign land, what "undiscovere'd country, from whose bourn / no traveller returns" the paddy is headed. His being left behind alone is the body left behind after the soul departs.
So there are plenty of interpretations; no doubt other readers could come up with several more just as convincing (or otherwise). Nor are these interpretations mutually exclusive, as they all could be operative at once.
So it's never a great idea to try to pin down the exact meaning of a poem. As Archibald MacLeish wrote:
A poem should not mean
Asking exactly why the narrator is left behind is a bit beside the point; he's left behind because that's how the poem works.