The ten days at the beginning of the Jewish month of Tishrei—from the Jewish New Year, Rosh Hashanah, to the Day of Atonement, Yom Kippur—are known as the “Aseret Yemei Teshuvah,” or the “Ten Days of Repentance” (more accurately translated as the “Ten Days of Return”).
When the prophet Isaiah tells the Jewish nation to repent for their sins, he says (55:6), “Seek G‑d when He may be found; call Him when He is close.”
Rabbah the son of Abuhah would say, "These are the days between the holidays of Rosh Hashanah and Yom Kippur."
The above serves as the basis for Maimonides’ explanation:
Despite the fact that repentance and crying out to G‑d are always timely, during the ten days between Rosh Hashanah and Yom Kippur it is exceedingly appropriate, and is accepted immediately.
Rabbi Shmuel Eliezer Edeles (1555-1631), the Maharsha, explains in his famed commentary on the Talmud that since the final judgment for the New Year is not signed until Yom Kippur, one still has the chance to beseech and return to G‑d during these days and change the judgment.
Our Sages in the Midrash explain that G‑d says about these days, “The gates of Heaven are open, and I will listen to your prayers.”
Based on this understanding of the Sages, Maimonides states:
For these reasons, it is customary for all of Israel to give profusely to charity, perform many good deeds, and be occupied with observance of G‑d’s commandments from Rosh Hashanah until Yom Kippur to a greater extent than during the remainder of the year.
Rabbi Moshe Isserles, the Ramah, writes in the Code of Jewish Law, “During these ten days of repentance, one needs to introspect and clean up his deeds and return to G‑d.”