She knows all sorts of proverbs.

Literal

She [topic-は] proverb [object-を] several [also-も] knows.

いくつも is a quantity expression: いくつ ('how many') + も ('also/even') = 'many, several, any number of.' Often paired with positive predicates to mean 'plenty,' it differs from 一つも ('not one') which only works in negatives. ことわざ is the native word for 'proverb, saying' — a category of fixed traditional expressions widely valued in Japanese culture, with thousands documented in collections like the Iroha Karuta. Knowing many proverbs is a marker of cultural literacy and verbal sophistication, often associated with older speakers or well-read individuals.