The “source” is from the Torah (Bereishis 24:65), where Rivka Imanu covered her face when meeting Yitzchok Avinu. It is also brought down in the Gemorah (Ketubos 17b, Rashi ad loc).
Halachicly, the Tur says that the Choson covered the Kallah with an article of clothing. that was the legal chuppah act. The Tosafists write that the covering of the bride’s face with a veil finalized the marriage. The Bach rules that we perform the Badeken and all the other shittos (canopy) of what constitutes the legal marriage in order to cover all halachic possibilities. The bride is veiled, and the overhead canopy is the groom’s covering for the bride.
The veil emphasizes that the groom is not solely interested in the bride’s external beauty, which fades with time; but rather in her inner beauty which she will never lose.
The veiling also symbolizes the bride’s commitment from this moment on to reserve her beauty for her husband’s eyes (tznius).