。
The other day he told me he'd lend me this book tomorrow.
Literal
Tomorrow this book [object-を] you [to-に] lend give [volitional-よう] [quotative-と] other-day he [topic-は] me [to-に] said.
A long sentence with reported speech. 貸してあげよう = 'let me lend' — the volitional of 貸してあげる ('lend as a favor'). The whole quoted part ends with volitional, then と marks it as quoted speech attached to 言った. Note the surprising word order: the main clause comes at the end, after the reported content. Japanese's ability to front reported speech this way is structurally different from English. Also note: 君に貸して挙げよう uses 挙げる (atypical kanji for あげる here — more commonly written 上げる or purely hiragana).