She writes me a letter once a month.

Literal

She [topic-は] month [per-に] one-time me [target-に] letter [object-を] gives-to-me.

Two grammar threads. First, frequency: 月に一度 = 'once a month' uses the per-unit に — the same に in 週に三回 ('three times a week') or 一日に二時間 ('two hours a day'). 一度 itself is a counter phrase ('one time'). Second, the giving verb くれる is the inward-facing benefactive — used when the gift comes 'toward me / our side.' Switching to 送る ('to send') would lose the warm, gift-flavored framing; くれる makes it sound like a personal gesture rather than mere correspondence. The ~に before 私 marks the recipient.