In this poetry, there are kakekotoba or pivot words: “mino” is used as a rhetorical device. First on the literal level “mi” in “mi no/実の” means fruit, then on subsidiary homophonic level “mino/蓑” means straw raincoat. According to the explanation of his poetry, the prince said that….

When I was in his villa, someone came to rent a straw raincoat (mino).
Instead of lending a raincoat, I handed a branch of yamabuki.
The following day, that person came to ask the meaning of that.
So, I composed a poem and give him it.

It means….
There are many beautiful double-flowered yamabuki blossoms in my garden.
But, I am sorry, I don’t have any raincoat (mino) just like yamabuki have no fruit (mi)
That is to say, Prince Kaneakira asked the visitor a sophisticated riddle. This poetry was very cool and future generations have loved it.

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Here is also very famous episode about Ota Dokan 太田道灌(1432-1486), a popular general in the Muromachi Period.

One day, he went on a hunting trip and was caught in a sudden shower.
He rushed into a shabby dwelling and shouted for someone to lend him a straw raincoat.
A young girl appeared and gave him a branch of yamabuki without saying anything.
Dokan got angry and returned home in the rain.
During the night, he told the story to his followers.
He told the story of the poetry of Prince Kaneakira and inferred that the girl compared poverty to the blossom of the yamabuki and wanted to say…
“I am so sorry, but I am poor and don’t have a raincoat.”
Dokan was shocked to hear that and ashamed by his lack of wit.
After that he started to learn waka poetry.

Text from Kikuko’s website: yamabuk