She grew increasingly beautiful as she got older.

Literal

She [topic-は] big becoming [as-につれて], more-and-more beautiful became.

Two grammar points. ~につれて ('as / in proportion as') is a correlation marker — 'as X happens, Y happens accordingly.' It implies a parallel progression of two changes. Attaches to the dictionary form. ますます ('more and more, increasingly') is a reduplicated adverb of progression, often paired with comparative or change-of-state expressions. The combination ~につれて + ますます is a natural collocation for tracking gradual change. 大きくなる here means 'grow up' — the everyday phrase for becoming an adult.