This is a cute wordplay by Basho but the wordplay can’t be translated into English. One word for rainy season (June-July) in Japan is “tsuyu” or 梅雨 in the kanji. Kanji can be pronounced in different ways, depending on whether they are used individually or in combination with other kanji. When 梅雨 is separated into梅 and 雨, it becomes “ume” and “ame,” meaning Japanese sour plumbs and rain. So he is playing with the word “rainy season” by breaking it into its kanji components of “sour plumb” and “rain” and saying that the sound of rain makes his ears sour. Another note is the rhyming of “ume” and “ame.”

With the falling sound

My ears also sour—

Of plumb-hinted rain

Furu oto ya

Mimi mo suu naru

Ume no ame

Falling sound and

Ears also sour become

Plumb’s rain