'Producing' is the right form of the verb. There is no mistake. This is the correct sentence: "Biennial plants last two years, producing flowers in the second year." 'Producing' is a present participle, and 'producing flowers in the second year' is a dependent participle clause referring to 'biennial plants'. You cannot use 'produce' here. For 'produce' to be correct, you would have to add a separate verb phrase, for example: "Biennial plants last two years, and [they] produce flowers in the second year."

Biennial plants live for two years, blooms/flowers in the second year and then dies.