There are some references to the season in the text of the play. Epiphany or Twelfth Night was quite an important festival at the time. You remember the part where Toby Belch complains about Malvolio, because he thinks he is virtuous there will be no more cakes and ale. Sir Toby is suggesting that it is fitting to dance, drink and celebrate for the festival. The festival setting gives a good excuse for lots of songs and dancing in the play, too, which were popular in the theatres of Shakespeare's time.
At the time when Shakespeare was writing this play, lots of the religious festivals were being controlled and restricted because of the religious and political changes in England (festivals were regarded as being too 'Catholic' for good Protestant Englishmen.) Some people felt that they were losing some good festivals and reasons for celebration because of too much reforming puritanical spirit. Sir Toby's punishment of Malvolio is partly motivated by this.