Dakshina is one of the four Vedic goddesses of Knowledge, the other three being Ila, Sarama, Saraswati. Ila is Truth vision, Saraswati is Truth audition, Sarama is Intuition and Dakshina gives right discernment. Her association to giving with regard to teachers came up because in ancient India knowledge was not supposed to be sold. While there were Gurukula supported by the king where every child could receive education, specialised knowledge such as Brahmgyan (knowledge of the Ultimate Reality or the Highest Knowledge) as well as certain forms of Shastra Vidya (advanced weaponry) was only imparted to those whom the teacher felt right through adhikar bhed (right discernment as to who should receive). There was no money transaction for this knowledge since no price tag could be put on it. Hence when the student had received all that he could receive, as a token of gratitude, he had to offer what the guru asked applying once again his right discernment keeping in mind the capability of the student.
In the setting of a yagna its esoteric sense means that the purification of nature through the yagna has not resulted in the right discernment then it is not yet complete. While Ila, Saraswati, Sarama can grant to the aspirant of yoga the higher faculties through which he can access a higher knowledge, this higher knowledge must be translated into the right action (daksha or perfect skill and mastery in action) through the application of the right discernment. Else the knowledge remains largely theoretical though true. Dakshina steps in to complete it by giving us the right knowledge for detailed application.