My father told me to write a reply and mail it right away.

Literal

Reply [object-を] write, immediately mail [command-せよ] [quotative-と] father [topic-は] me [to-に] said.

The classical imperative せよ (from する) appears inside quoted speech — father's direct command was 投函せよ. This archaic-sounding form survives in quotes, formal directives, and set phrases. 投函する specifically means to post a letter by dropping it into a mailbox. The て in 書いて chains two sequential actions: write, then mail.