Because of her mother's illness, she had to stay home yesterday.
Literal
Mother [gen-の] illness [because-of-のため] she [topic-は] yesterday home [at-に] had-to-stay.
~のため is a 'because of' / 'due to' causal connector, slightly more formal and written than ~から or ~ので. The subject 彼女 is marked with は, and 居る ('stay/exist, animate') takes に for location. The double negative なければならなかった is the past tense of 'had to' — 'the state of must' transposed into the past.