Shemini Atzeret: "the pause of the 8th day." The day after Sukkot, when the Holy Blessed One murmurs in our ear, "Don't go just yet. Don't leave this bower where we've spent this past week. Linger a little longer with Me."
Shemini Atzeret: the day when we shift, in our daily amidah, from blessing God Who brings the dew to blessing God Who brings the wind and the rain. (And in this climate zone, eventually, the ice and the snow.) On this day we recite a special prayer for rain.
Shemini Atzeret: another opportunity to say the prayers of Yizkor, the liturgy of remembrance which recite as we remember our beloved dead. Of course, many of us just said Yizkor. There was a Yizkor service on Yom Kippur, a scant twelve days ago. Why are we reciting Yizkor again so soon?
Well. Like anything else in Judaism, there is a multiplicity of answers. But here's an answer which speaks to me.