The Chida (in his sefer Nachel Eshkol) writes that the roshei teives of the pasuk: ישא אביוניםלמתנות ורעהו ל are אלול. This is to teach us, that we should:ה' להתערר ליראה את כשם שמתעורר בחודש אלול – “Work on fearing Hashem, the same way we do during Elul”.
We find a remez to this idea in the Mishnayos in Rosh Hashanah (3:9). The Mishnah says: If one passes by a shul, or he lives near the shul and he hears the sound of the shofar or the sound of the Megillah, if he has intention to fulfill the mitzvah he is yoitsa. The Mishnah mention both shofar and Megillah together, to hint at the fact that the mitzvos of shofar and Megillah are related.
ומי שיש לו מוחו בקדקדו ישפוט בצדק כי בעבור שהגדיל ה' עמנו מאוד להצילנו ממות לחיים אין ראוי לאבד ולהפר יראה ולפרוק עול תורה ומצוות חם ושלום וגילו ברעדה כתיב ודי למי שיש בזה לב - “One who has a brain in his head, will think about how Hashem saved us from death, and gave us life. He will make sure not to waste this opportunity, and won’t treat Torah and mitzvos lightly, one who has a heart will head this warning”.
The Tikunei Zohar (p.57b) describes the greatness of Yom Kippur, referring to is as a day like Purim. This is the reason Yom Kippur is called: יום כ-פורים , translated as “A day like Purim”.
The Mishnah Berurah (90:28) quotes a Teshuvos Radvaz that discusses a prisoner whose jailors permit him to daven with a minyan for one day. The question is, which day should he choose? Should he request to go to shul for Yom Kippur or rather on Purim? The Radvaz advises that he go to shul at the first opportunity because if he pushes off the offer, one will know what will be. Therefore, right away the following day, he should go to shul to daven.
Nevertheless, when we study his question, we notice the importance of specifically these two days, Purim and Yom Kippur. He didn’t even write Rosh Hashanah. Purim and Yom Kippur are at the top of the list of days we want to be in the beis medresh.