。
You can check out up to 5 books at a time.
Literal
Book [attributive-の] lending [topic-は] 1-time [at-に] 5-books [up-to-まで] [is-です]-[insistence-よ].
貸出し is the noun form of 貸し出す ('lend out'). 1度に ('at one time, at once') is an idiomatic time expression. まで ('until, up to') here marks the upper limit. The counter for books is 冊 (さつ). A typical library-rules statement, with よ softening it to a friendly heads-up.