The defining fact of Chai Elul as a Chassidic yom tov is that it is the birthday of both the Baal Shem Tov and the Alter Rebbe: Likkutei Sichos 24 (Chai Elul) states explicitly that the Baal Shem Tov was born on Chai Elul 5458 (a Monday) and the Alter Rebbe on Chai Elul 5505 (a Wednesday), making the day a double occasion of foundational significance for Chassidus.
An ancient Chassidic saying, preserved in two versions in Likkutei Sichos 29 (Chai Elul), captures the day's essential character: one version holds that Chai Elul 'brought and continues to bring vitality into the month of Elul,' while the other holds that it 'gives vitality to the spiritual labor of Ani L'dodi V'dodi Li' — and, on our reading of that cited page, the Rebbe explains these as two complementary paths, the first associated with the Baal Shem Tov's mode of divine revelation from above, and the second with the Alter Rebbe's emphasis on personal inner effort from below.
The deeper meaning of this dual birthday is explored in Likkutei Sichos 24 (Chai Elul), which concludes, on our reading of the cited page, that Chai Elul represents the 'birth' of Jewish spiritual life from a passive, fetus-like state into an active 'living soul,' achieved by drawing Chassidic light into the vessels of intellect and daily action, and that this process of integration ultimately hastens the redemption.
The relationship between the two birthday figures is articulated in Likkutei Sichos 19 (Chai Elul), which records the formulation of the Previous Rebbe: the Baal Shem Tov showed 'how one must serve G-d,' while the Alter Rebbe showed 'how it is possible to serve G-d' — general Chassidus inspiring the soul with the will and faith to serve, and Chabad providing the intellectual tools to internalize and execute that service in practice.
Likkutei Sichos 4 (Chai Elul) deepens this connection by noting that since the Alter Rebbe received the innermost dimension of the Baal Shem Tov's Torah, he thereby received the Baal Shem Tov's innermost essence itself — and, on our reading of the cited page, the Rebbe concludes that this is why the Alter Rebbe is considered the Baal Shem Tov's true 'grandson,' the one who integrated that essential spiritual light and channeled it into human comprehension through Chabad Chassidus.