Each festival was established for a different reason, but there is a common aspect to all of the holidays: It is prohibited to perform labor on these days. Why? So that families have time to be together, and so that rabbis are able to speak to the entire community. The festivals have a purpose, and the cessation of work prevents us from becoming distracted from this purpose.

Not working allows people to pay attention to the essence of the day:

The reasons for this mitzva [of refraining from labor on holidays] include: For Israel to remember the great miracles that God did for them and their patriarchs, and for them to speak of them, and make them known to their children and to their children’s children. Resting from the affairs of the world will free them to engage in this. If work, even light work, were permitted, each and every person would turn to his affairs, and the honor of the festival would be forgotten from both children and adults.

Free time is dedicated to Torah study:

Furthermore, there are many advantages to cessation of labor, as the entire nation gathers in the synagogues and study halls to hear the words of the Torah, and the leaders of the nation guide and instruct them, as the Sages said: “Moses instituted for the Jewish people that they should expound the halakhot of Passover on Passover, and the halakhot of Shavuot on Shavuot” (Megilla 32a). (Sefer HaĤinnukh 298)

Further reading: For more on the cessation of labor as an opportunity to gather for communal learning, see p. 35; A Concise Guide to the Sages, p. 125.

The holidays are not only opportunities for families and communities to gather together; they are holy days, when everyone assembles to pray and to praise God. Nevertheless, we do not stay in the synagogue the whole holiday; we must rejoice in our homes as well, with good meals and with concern for those who do not have enough to eat.

The reason that [the holidays are called] “holy convocations,” is because on this day everyone convokes to sanctify it. For it is a mitzva given to Israel to gather in the House of God on the day of the festival, to sanctify the day in public, with prayer and praise of God, in clean garments, and to make it a day of feasting. As it says in the Writings: “Go, eat rich foods and drink sweet beverages, and send gifts to whomever does not have anything prepared, as this day is holy to our Lord. Do not be saddened, as joy in the Lord is your strength” (Nehemiah 8:10). (Ramban, Leviticus 23:2)