'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."