Why is Brabantio so dismayed about Desdemona's marriage, despite the fact that Othello is so honored and admired in Othello?
It is one thing to appreciate a person's qualities from afar (or maybe not so afar, since Brabantio had received Othello as an honored guest in his home), but quite another matter to accept that person as a member of one's family. By displaying this contradictory trait in Brabantio, Shakespeare is skillfully holding the mirror up to all of us, not just citizens of 16th century England. We, in our modern society, still struggle with this sort of "two-faced" prejudice today.
That said, Brabantio not only decries the marriage (since it was completely improper for a daughter to elope without securing her father's blessing and permission), but goes further and accuses Othello of bewitching his daughter in order to get her to marry him:
She is abus'd, stol'n from me and corrupted
By spells and medicines, bought of mountebanks,
For nature so preposterously to err...
Sans witchcraft could not.
...I therefore vouch again,
That some mixture powerful o'er the blood,
Or with some dram conjur'd to this effect,
He wrought upon her.
Certainly, Brabantio is making reference to the impossibility of his daughter falling in love with a black man. But it is the witchcraft that he is objecting to before the Duke.
It is up to Othello and Desdemona to convince the Duke that they are married for love, not by the benefits of witchcraft and potions. And Othello's speech to the Duke, explaining the real situation of love between himself and Desdemona, is one of the most powerful and beautiful in all of Shakespeare (Act I, Scene iii, lines 128 - 170).
Brabantio is angry and dismayed at Desdamona's marriage because she has married without permission and the man she has chosen is old and black and not a Christian.
400 years ago, daughters were supposed to do what their fathers told them until they were married and then they should do what their husbands told them. Women were supposed to be obedient. It was the father's choice to allow or disallow his daughter to marry and the daughter was supposed to obey.
For rich families, a marriage was a chance to make an alliance with another important family. Love was a very minor concern. Daughters, especially beautiful, virginal daughters of rich merchants, were a very valuable and scarce commodity representing 20 years of investment. And this precious investment, Desdamona, has been stolen by a totally-unsuitable, socially-unacceptable husband in the middle of the night. It is a disaster for Brabantio's reputation and plans.
In the eyes of 16th C. Venice Brabantio's grand-children will be comi-tragic half-breed heathens and his daughter will be a walking scandal, an advert for his lack of domestic authority. She has made him look very foolish, cost him a potential fortune and risked the acceptability of the familiy's future generations.
It is a major, major social scandal. You could compare it to the fuss and gossip made about Paris Hilton and her jail sentence or Britney Spear's many mis-adventures.
Brabantio faces 'death by gossip'! He will be infamous as the man whose wild, lustful daughter ran off with a "_________" (insert various horrible names for muslims/africans here). His enemies at court will make sure this story is big news. They won’t let it be forgotten. It is juicy gossip. You know what people are like...
“Hey did you hear about Brabantio’s daughter? She ran off with a Moor!!! It’s disgusting, I’d never let my daughter behave like that.”
“It shouldn’t be allowed. Good, nice Christian white girls being ‘used’ by filthy men like him. Somebody should stop it.”
“There must be something wrong with the girl, maybe the family is not quite right. The apple doesn’t fall far from the tree, hmmm?”
"The children will go to hell as punishment."
Imagine when Brabantio is in any important meeting and one of his enemies says, “How’s your beautiful daughter?” with a tiny little smirk.
Etc. Etc... Death by horrid, vicious, hurtful, ignorant gossip.