Why does the Kallah wear a veil under the Chupah?
(and why is it generally transparent?)
I think the transparency is just a matter of modern convenience, so she can better see where she's going. Assuming the groom put the veil on her, and her whereabouts are known from that point (bedeken) until the chupah ceremony, we can assume it's the same woman.
The groom's act of placing a veil upon his bride is a very old, strong custom (I think Rosh?). One reason is to avoid Rachel/Leah switcheroos (see above). I think I've also heard that according to one Rishon (Tosafot - Yuma 13B - D"H "לחדא אמר לה על מנת שאכנס כו'"), the bedeken actually serves as nisuin; and/or that it symbolizes his obligation to clothe her.
Many have to custom to specifically use a thick, opaque veil, so that one can not see the bride's face. (See Nitei Gavriel Nisuin I, Chapter 13, halacha 9).
This veil is placed prior to the Chuppah, and is then worn until after the Chuppah. (see footnote 14 and here). One of the reasons given is so that the bride not see the ring she is being betrothed with, lest she mistakenly under or overestimate its value, causing the "transaction" to be based on an erroneous assumption - from here. - See Remah Even HaEzer 31:2
The She'erit Yaakov and Shulchan Ha'Ezer (quoted in footnote 14 of the Nitei Gavriel) say that one should not see the face of the bride during the wedding ceremony.
An esoteric reason for covering the bride's face with an opaque veil is given here. There, it is explained that just as the holiness radiating off Moshe's face after he descended Har Sinai required him to cover his face with a veil, the holiness that is radiated off the brides face during the ceremony needs to be covered as well.
My own conjecture here is that it is worn for modesty reasons based on Rashi to Shemos 26:9:
אל מול פני האהל: חצי רחבה היה תלוי וכפול על המסך שבמזרח כנגד הפתח, דומה לכלה צנועה המכוסה בצעיף על פניה:
before the front of the tent: Half its width [of the sixth curtain] was hanging and folded over the screen on the east[ern side of the Mishkan], before the entrance, resembling a modest bride whose face is covered with a veil.
upon receiving a gift that one truly longed for, the gorgeous wrapping on the outside makes it even more desirable.
as we learn from hagada gd implaced all the precious gems upon hava before bringing her forth to adam.
Leah represents Yaakov's Mazal (fate). She is the woman whom Yaakov was destined to marry. Rachel represents choice. She is the woman whom Yaakov loved and chose to marry. The narrative portions of the Torah contain much more than simple stories.
There is a profound message being taught here. When one gets married, although he thinks that he is marrying a Rachel, the one that he chose, there are bound to be unforeseen surprises. One will discover, after the wedding that he has also married a Leah, who represents the unanticipated nature of one's spouse. This unforeseen nature, however, is exactly what one needs in a spouse. As the song goes, "You can't always get what you want, but get what you need."
When the groom veils his bride, he is stating in effect, "I will love, cherish and respect not only the 'you' which is revealed to me, but also those aspects of your personality that are hidden from me. As I am bonding with you in marriage, I am creating a space within me for the totality of your entire being, including what remains veiled."