People always say "What's this holiday about anyway?". And the people who ask this question are people who keep two days of Shemini Atzeret (whether or not they know that this is what they're doing), and are asking about the first day of Shemini Atzeret. They know what the second day is about.
Both Shemini Atzeret and Shavuot (the other "atzeret") are biblical pilgrimage holidays tied to the agricultural cycle, with no specific biblical mitzvot surviving into post-Temple Judaism. Both have been transformed into holidays about Torah. In both cases, the Torah-related customs are concentrated on a single day in communities that observe those holidays for two days, while the other day is left mostly as generic yom tov with a few exceptions (Ruth, geshem). But the Torah-related day for Shavuot is the first day, while it is of course the second day of Shemini Atzeret. And this asymmetry means that people don't ask "What's the second day of Shavuot about anyway?" (Well, they do, but for different reasons. That's more procedural than spiritual.) ER suggests that the order is important to how people view the holiday. It's easy to see the 2nd day of Shavuot as a continuation of the first day. It's harder to see the 1st day of Shemini Atzeret as a continuation of something that hasn't happened yet. Also, the fact that the 2nd day of Shemini Atzeret gets its own label may influence perceptions significantly.
Another question asked around that time is why Kohelet is read during Sukkot (or Shemini Atzeret, depending on the year and one's minhag). In addition to all the other suggested reasons, I would point to the beginning of chapter 12:
So appreciate your vigor [or "remember your Creator"] in the days of your youth, before those days of sorrow come and those years arrive of which you will say 'I have no pleasure in them'; before sun and light and moon and stars go dark, and the clouds come back again after the rain.
And it goes on with many poetic metaphors. This is what Sukkot and Shemini Atzeret are about: a final opportunity to appreciate and rejoice before everything goes dark and rainy and cloudy.