I remember doing it, I was so freaking nervous.
I lived 3 states away and had to sneak into town to ask the father without my gf knowing I was in town. She still lived at home (well, had recently moved back home b/c I told her not to renew her lease). I took 2 days off of work (new job) and drove straight to a local restaurant where we were supposed to meet.
I picked someplace that neither of us would normally go, b/c we would have likely seen at least someone we knew, and word tends to spread...
He met me at the door, smiled, shook my hand and knew EXACTLY what I was doing. I simply said, very formally, I would like to ask Johanna for her hand in marriage, and I wanted your blessing before doing so. He laughed, punched my shoulder and we joked the rest of the meal. We met for breakfast, so there were no drinks.
My wife is pretty independent, liberal, what have you... Anyway, after I proposed and was telling her about the meal that I had eaten with her father, she just rolled her eyes, called us dorks and said we were so stupid. Much like she would have done if watching us cheer for our favorite football team.
IMHO, it is a harmless custom that means a lot to both the father and the son-to-be. It really helped us get to know each other better, and improved our relationship.
I think many girls' parents are hesitant to trust, embrace, get too close to boyfriends prior to knowing their intentions. All this custom really amounts to is letting her closest family members know of your intentions.
As the father of a daughter, I know that at 25, I will still feel responsible for my daughter. Not that I will try to run her life, or make her decisions, but i will want to know she is okay. She is only 3 now, so that might change, but I think it is just respect paid to the family.
As many people have said, you DO, in fact, marry into a family. No truer words have ever been spoken.