Joy of Sukkos follows purification of the heart

זמן שמחתינו.

"The time of our rejoicing" (zman simchaseinu).

The Sefas Emes opens with the description of Sukkos as the festival designated as the time of our joy.

דכתיב לישרי לב שמחה.

As it is written, "For the upright of heart, joy" (Tehillim 97:11).

The verse teaches that genuine simchah belongs specifically to those whose hearts are straight and upright.

לכן אחר ר"ה ויוה"כ שנעשין בנ"י ישרי לב.

Therefore, after Rosh Hashanah and Yom Kippur, when Bnei Yisrael become upright of heart.

The Days of Awe purify Bnei Yisrael and restore them to an upright, straightened heart - which is why the joy of Sukkos follows precisely then.

דכתיב האלקים עשה את האדם ישר.

As it is written, "Hashem made man upright (yashar)."

Man was originally created straight and upright; uprightness of heart is his true, natural state.

ורק ע"י לכלוך העונות חשבונות רבים.

And it is only through the soiling of sins that "[they sought out] many calculations."

The continuation of that verse - "but they sought many calculations" - describes how aveiros soil a person and twist his originally straight heart into crookedness and complication.

ולכן אחר ימי הטהרה מתעורר השמחה:

And therefore, after the days of purification (yemei ha'taharah), the joy is awakened.

Once the cleansing days of Rosh Hashanah and Yom Kippur have washed away the soiling of sin and restored the heart's uprightness, the true simchah of Sukkos can finally be aroused.

Summary: Sukkos is "the time of our rejoicing" because joy is reserved for "the upright of heart." Man was created yashar, but sin soils him with "many calculations." After the purifying days of Rosh Hashanah and Yom Kippur restore Bnei Yisrael to their original uprightness, the genuine simchah of Sukkos is awakened.