Ever go jogging? When you're done, you're not supposed to stop suddenly; it's too big of a shock to the system. Rather, you "warm-down" by walking, and finally you can stop.
The same is true of a Festival. We don't go directly from a joyous festival back into everyday life. We need a day to "warm-down," or rather to "wrap-up" the joy of the Holiday. "Isru chag" is that day. "Isru chag" literally means "tie up the festival." That is, take the happiness and meaningfulness of the festival and bring it with you into the rest of the year.
In Temple times, festive offerings were brought on isru chag of the Shavuot Festival. Isru chag remains a semi-festive day, even today when there is no Temple.
Sources:
- Shulchan Aruch 429:2
- Succah 45b
- Jerusalem Talmud, Avodah Zarah 1:5
