She can neither write nor read.

Literal

She [topic-は] write [nominalizer-こと] [subject-が] unable [and-ず], read [nominalizer-こと] [also-も] unable.

できず uses the classical negative form ~ず (equivalent to ~ないで or ~なくて), giving the sentence a formal, written feel. The parallel structure — 書く事ができず、読む事もできない — lists two inabilities with the second emphasized by も (also). Note the kanji 事 used for the nominalizer こと, which is more common in older or more formal writing; modern style guides often prefer hiragana こと for grammatical uses.