Having a scribe (sofer) inspect the parchment of the mezuzah before Rosh Hashanah is a common custom (even though, according to most rabbis, mezuzot need checking only once every three and half years).
The mezuzah symbolises God’s protection over Jewish homes. The parchment inside the case contains the first two sections of the Shema (Deuteronomy 6 and 11), which describe our commitment to God and the mitzvot and the blessings God will then grant us.
Rosh Hashanah is when we stand before God in judgment of the last year’s deeds and take stock of ourselves in preparation for a new and hopefully better year.
Checking the kashrut of our mezuzot — that no letters have become blurred or that the parchment has not suffered any damage — is an expression of our yearning to put our spiritual lives in order and to renew God’s protection in our lives.