The bee represents Mary, the mother of Jesus. (In fact, it's a fairly safe bet that just about anything represents Jesus or Mary or both.) Other versions of the Exultet text make this explicit; the "Franco-Roman version" has a long we-love-bees section, as recorded by Thomas Forrest Kelley in his The Exultet in Southern Italy (OUP, 1996), p38, concluding with:
O vere mirabilis apis cuius nec sexum masculi violant, fetus non quassat nec filii destruunt castitatem. Sicut sancta concepit virgo Maria, virgo peperit et virgo permansit.
O truly marvelous bee, whose sex is not violated by the male, nor shattered by childbearing, neither do children destroy her chastity. Just as holy Mary conceived as a virgin, gave birth as a virgin, and remained a virgin.
This alludes to the old belief that bees reproduced asexually. Isidore reports in Etymologiae 12.8 that bees, like other small flying animals, are spontaneously generated from the carcasses of animals. So also, Augustine says in City of God 15.27 that bees have no distinction of male and female (alia vero in quibus nihil sit maris et feminae, sicut apes). This comes from ancient sources: Pliny in Natural History 11.16 suggests that they might weave their young out of parts of flowers; and Aristotle (On the generation of animals 2.10) says they are hermaphrodites, and do not copulate with one another.
We now know that they really couldn't have been more wrong, but the sentiment is there.
We have several copies of "Exultet rolls" which have bee illustrations to accompany the vigil text, indicating the great interest and respect that people had for them. Here is an eleventh century one, from Bari (Archivio del Capitolo Metropolitano, Exultet 1).
Bees were an example of industry, and esteemed because they provided so many useful things - in particular, honey and light (via the candle). Both honey and light have obvious Biblical resonances. Additionally, gifted orators and writers were said to have a "honeyed tongue", and so some saints may be depicted with accompanying bees (e.g. Ambrose, Anthony of Padua, Bernard of Clairvaux, Isidore of Seville).