One common, powerful theme of the high holidays is the idea of the broken heart. This is encapsulated prominently in the blowing of the shofar, with its shevarim [shattered] call. (See, e.g., The Shofar as Prayer at My Jewish Learning.)
All who hear the ram’s horn — during the preparatory month of Elul and the Days of Awe — are meant to experience a broken heart. And, so according to this story, is the one who sounds the shofar:
Rabbi Wolf, shofar blower in the synagogue of the Baal Shem Tov, has been studying special intentions for his annual role, but loses his crib sheet on the bima; forgetting everything, he blows the shofar with a broken heart. The Baal Shem Tov tells him,
“In the Palace of the King there are many rooms and halls, and each door to a room or a hall has a different key. But there is a better way to enter than to use the key, and this is to use an ax, which can open the locks of all the doors. The same is true of proper intentions. They are the keys to each and every gate, and every opening has the proper intention for it. However, the broken heart is an ax. It allows every person to enter all the gates and the halls of the King of Kings, the Holy One, Blessed be He.”
— Moshe Chaim Kalman, Or Yesharim.
see also The Light and Fire of the Baal Shem Tov.
Yitzhak Buxbaum. NY: Continuum, 2006.
Broken-heartedness is often described as requisite for prayer, particularly at the Days of Awe, as in this teaching from Abraham of Slonim (19th Century CE):
You should act in prayer as if you were a farmer: first you plow, then you seed, afterward you water, and finally things begin to grow. In prayer, first you have to dig deeply to open your heart, then you place the words of prayer in your heart, then you allow your heart to cry.
— found in Machzor Lev Shalem

In a season dedicated to atonement and forgiveness, reminders to open one’s heart are important. But how are we meant to respond to this call when our hearts are already broken? when we’re just barely hanging on?
Around the corner from the temporary synagogue where Fabrangen Havurah holds high holiday services, this message was painted on the sidewalk.
…An alternative thought for Shabbat Shuvah [the sabbath of “return”].