Learn Farsi: مهر ماهِ پاییز است.

مهر (Mehr) is the seventh month of the Iranian solar calendar. Because the Iranian calendar aligns perfectly with the solar seasons, مهر always marks the exact beginning of autumn, corresponding roughly to late September through late October.

Why is there an "e" sound connecting ماه (māh) and پاییز (pāyiz) when this is spoken?

This is the Ezafe particle, which is used to link words together (usually a noun to its modifier or owner). While the English translation uses a noun adjunct ("an autumn month"), Farsi links the noun ماه (month) and the noun پاییز (autumn) to literally say "month of autumn" (ماهِ پاییز).

The English translation says "an autumn month", but why isn't there a word for "a" or "an" in the Farsi sentence?

When making a general statement of classification—like defining what category something belongs to—Farsi often omits the indefinite article. Saying مهر ماهِ پاییز است is the standard, natural way to express this fact, without needing to add یک (one/a) or the -ی suffix.