Why do we continue to keep two days of Yom Tov outside of Israel? The reason for two days was revelant during the temple era as we needed two witnesses to see the new moon. But today, with no Temple for witnesses to come too and a fixed calendar which tells us when the new moon is for hundreds of years, keeping the second day seems like an unnecessary hardship.
Aside from the reasons that Shalom mentioned, the Midrash (Shir Hashirim Rabbah 1:6) states that it's part of the penalty of exile: since we didn't properly keep the one day of Yom Tov in the Land of Israel, we now have to keep two in exile.
(In a more positive vein, though, Chassidic writings see this as part of the process of teshuvah - like a rope that was severed and now is re-knotted, where the part with the knot is much thicker than the rest of the rope. In the same way, teshuvah involves an even greater increase in Jewish practice than before - including, in this case, twice as much Yom Tov.)
On page 31 of Rabbi Zev Reichman's book based on the shiurim of Rav Moshe Wolfson Shlit"a, Flames of Faith: An Introduction to Chassidic Thought, he quotes Rabbi Isaac Luria (the holy ARI) who says the reason for a second day of yom tov is because anyone who lives inside the land of Israel has an extra soul, and therefore can internalize the holiness of a festival in a single day. With only half the soul-power, those who dwell outside Israel would find it impossible to internalize the blessings of the blessings of the holiday if they only have half the spiritual capabilities prescribed as ideal in the Torah.
The Babylonian Talmud just says "well, that's our ancestors' custom, so we keep it." Keep in mind that this source is more interested in HOW the practice came about, rather than WHY.
It's been suggested (I believe in the Yerushalmi Talmud) that this arrangement works out because the Jews of Israel can fit enough spirituality into one day of Yom Tov; the Diaspora Jews couldn't, so they get two days to try.
One of the medieval commentaries (I believe the Ritva) has the well-known suggestion that the 2-day rule was in fact intended to encourage people to move to Israel, and thus to penalize those who didn't!